Here are the problems that stand out immediately:
1. DNA delivery
The first issue with making animal/human hybrids out of humans would be to deliver the DNA to every cell in the body. It is estimated that the human body has somewhere around 30 trillion cells. Thus, the first problem would be to somehow deliver the modified DNA into such a great number of cells. Let us assume for the sake of argument that you have a virus that contains the entirety of the DNA that needs to be introduced into the target cell, along with the DNA coding for the enzymes needed to introduce it into the cell, and that this does not produce any adverse effects. Even then, it would take an extremely long time to actually infect all of the cells (Or an acceptable amount thereof). For instance, the time it takes for an infection with HIV to progress to AIDS is stated to be around a decade, and there are much less cells involved in this. If your objective is to only modify certain characteristics (Such as increasing muscle mass by disabling myostatin production), you might get away with infecting only certain cells and accepting the issues involved. If your organization is not content with waiting over a decade for enough of the host's cells to be infected by the virus after an injection, you could have part of the transformation process involve the 𝚟̶𝚒̶𝚌̶𝚝̶𝚒̶𝚖̶ patient being hooked up to an IV drip of fluid that contains countless virions.
2. DNA Delivery Part Two
Presumably, the means of delivering the modified DNA that is to be introduced into the host, along with DNA coding for the enzymes to do so, would be done by a virus. Assuming that you do intend to modify the human body completely, you would need a virus that is capable of infecting all types of cells (or several viruses, each capable of infecting a particular cell type but all of them still modifying the same genes in the same way, but, in both cases, there is still the issue of immune response). The first thing that is necessary for a virus to enter a host cell is for a receptor on the surface of the virus particle to bind to a receptor on the host cell. Thus, in order to use a virus to deliver the modified DNA, you must either have a virus that can bind a receptor that is found on every type of cell (I don't think such a receptor protein even exists, but I could be and probably am wrong), or use several different viruses, each capable of binding a receptor found on a particular cell type.
3. Viral interference with bodily functions
Another issue that is likely to arise with the use of viruses for DNA modification is the interference of the virions themselves with things in the body that REALLY shouldn't be interfered with. As an example, the spike protein on SARS-CoV-2 has been found to bind to fibrinogen and induce blood clots. If, for instance, the life cycle of the virus used to enact the genetic modification disrupts cell function, or the proteins on the surface of the virus are capable of acting on cell signaling pathways, there is a good chance that the host is going to die.
4. Immune Response
As you are aware, drastically modifying host cells is going to trigger an immune response. One such means of doing that is that MHC molecules on the surface of a cell display fragments of proteins that are produced and degraded by the cell to immune cells. If the antigens on display are something that is not recognized by the immune cells as part of the body, the immune system will attack. Unfortunately, some of the protein fragments that are going to be on display would be parts of viral proteins and will get recognized as foreign. However, this may actually be a solvable problem. There are some studies that investigate transplanting umbilical cord blood stem cells instead of bone marrow cells, since UCB cells are not yet "trained" to recognize which antigens are supposed to be present and which are not, as a means of preventing graft-versus host disease. So, the use of either gene therapy or transplantation of the appropriate cells might be used to solve this issue in the host.
5. Physical Development
This is probably the biggest issue with transforming someone into a human/animal hybrid. For instance, replacing the DNA of a human with that of a dog will not result in that person turning into a dog, even if you could magically keep that person from dying. A lot of things, such as bone structure, are determined during embryonic development. Chemicals known as morphogens are produced by embryonic cells and diffuse throughout the tissues of the developing fetus. The concentrations of these chemicals regulate gene expression and cell differentiation in the embryonic cells, which in turn determines things such as body plan and how the organs are arranged. As a sidenote, this is thought to be why accutane causes birth defects: The accutane molecule resembles one of the morphogens enough to act as a morphogen and disrupt fetal development.
Furthermore, post-embryonic development must also be properly regulated. For example, bone growth takes place at a plate of cartilage at the end of a person's bone called a growth plate. To put it simply, some cartilage at the end of a bone gets turned into more bone, but the cartilage cells also divide to produce more cartilage cells. During puberty, the rate at which cartilage is replaced by bone at the growth plates is greater than the rate at which new cartilage is produced, and eventually all the the cartilage is replaced by bone, which is why people stop growing.
Your method of producing animal/human hybrids will have to somehow regulate the development of the new body plan from the human one (For instance, changing the shape of the skull, other bones, muscles, and various organs). If the physical development also involves temporary structures, you must also figure out how to introduce these structures to the host's body and remove them when necessary. I've roughly explained how it's done starting with an embryo. You would have to recreate that sort of biochemical signalling not only to break down the old body plan, but to generate the new one correctly. Development starting with a zygote has the advantage of not having to work with a developed body plan and having plenty of stem cells to turn into cells that are needed. Transforming a full-grown human is going to be much more complicated.
If you want your hybrid to be capable of reproduction, the modified genome would also have to be able to govern embryonic development correctly. I have no idea how these goals could possibly be achieved. Furthermore, the transformation process has to be survivable. If, as an example, the transformation gradually compresses the braincase, the patient is going to have a very bad time. Nevertheless, if you insist of transforming a full-grown human, the process would be very hands-on, probably with copious surgical intervention and other treatments. In that case, one has to question whether the cost of time, money, and RnD is really worth it to produce an expendable soldier and what advantages this would offer over alternatives.
6. Gene Expression
The process must ensure not only that the DNA of the host is modified, but that the correct cells express only the correct genes. For instance, you do not want a retinal cell to express genes for the production of insulin. Your method of genetic engineering would have to be capable of properly regulating gene expression in the host. Furthermore, you will likely need a means to disable certain genes at a certain point in time after the modification process.
7. Alternative Splicing
It is possible for one gene to code for multiple proteins. The RNA produced during transcription of a given gene can be spliced in different ways before translation, leading to different proteins with different functions. The overwhelming majority of human genes are thought to give rise to RNA that undergoes alternative splicing. Your process of genetic modification would have to address this if modifying genes which give rise to RNA that undergoes alternative splicing.
Other issues with the idea of genetically editing someone to produce human/animal hybrids have been discussed by the other answer, and some you are already aware of. My conclusion is that, as proposed, this is extremely implausible.
However, if your organization has access to the enormous computing power required to design the genome for an animal/human hybrid and simulate its development to ensure that all goes according to plan, you can have the organization create a zygote and grow the hybrid in an artificial womb (Seeing that the technology is being explored today, this is more plausible), as an example. This would mean that you do not have to worry about problems associated with transforming a full-grown human. You can either have the hybrid indoctrinated into serving whatever purpose the organization has for them, or transplant the brain of a willing subject into the hybrid body (And do whatever you want with the hybrid's original brain, you did say your organization has no ethics and no oversight). This path does present its own problems, though, namely that the organization would have to take care of the hybrid somehow until it matures (Or use accelerated aging handwavium) and the problems associated with transplanting a brain, plus the fact that the supply of hybrid would be limited (So, they wouldn't exactly be expendable).