The limitations upon the duration of gestation and maturation of placental mammals are not based solely upon the rate of cell division, they are based upon the rate at which the mother can supply nutrients to the fetus for gestation, and the rate at which the newborn can consume nutrients or its mother supply them, and additionally the amount of time that it takes the creature to learn to be a fully functioning member of its society.
Since a fetus grows at a constant rate, this requires a constant rate of nutrient and energy input, and if the demand for either is not met, it can lead to health problems for both mother and offspring.
Likewise, meeting a particular post-birth growth rate requires meeting the demand for energy and nutrients, or suffering from health issues. Additionally, many species have reduced growth rates due to the time required to learn the social aspects of being a member of its species.
For a bioengineered species, assuming no limitations on the capabilities of the designers, the social limitations on growth need not apply, since the necessary information can be built in at design-time.
Additionally, for a species designed to be a bioweapon, there is no need for it to gestate its own young if its young can gestate within the species that it is intended to exterminate. In fact, the act of implantation and gestation could be a primary mechanism by which the target species is exterminated, as well as acting as a self-limiting control upon its population.
However, if this bioweapon species is designed to target a sentient, technological species such as humans, relying upon humans as a reproductive host may not be completely reliable, given humans' capability to rapidly recognise the threat and act to minimise it. By necessity, a bioweapon designed to grow as rapidly as possible would have a somewhat different biochemistry, and the fact that such a difference exists means that it could be targeted by selective poisons that affect only the bioweapon and not the human host.
So... if we assume no limits on the availability of energy and nutrients, how rapidly could cells divide? Human cells divide at most once every 24 hours, though some yeasts can divide every 52 minutes, maintaining a constant maximal cell mass, while some fly embryonic cells have been timed at 8 minutes between divisions, though these would be trading mass for quantity, and would not be growing.
Most of the time in a steady size cell division cycle is taken up by growth, though a significant time is spent replicating DNA. in order to minimise the time spent replicating DNA, the DNA may be pared down to the barest minimum necessary, and split up into many seperate chromosomes, allowing the DNA to be replicated in parallel. It is not beyond the bounds of possibility for the bioweapon's cells to divide each 30 minutes. However, from an initial single cell, which may be very large, we could have a number of rapid divisions without growth, in order to achieve a larger number of smaller cells as quickly as possible, thus multiplying the growth rate of the organism in as small a time as possible.
If we take an initial cell of about 128ng, and rapidly divide it down to the average cell mass of the organism, that most likely being 1ng, that would be 7 divisions. These 7 divisions could conceivably be achieved in 5 minutes each, for a total of 35 minutes to reach the 128-cell stage. From there, with cell growth included, each cell division could in theory be achieved in as little as 30 minutes, though in practice, it would take somewhat longer due to cell death: not all cells survive (or are intended to survive), so it is reasonable to say that we could have a doubling in mass each 40 minutes, conditions permitting.
That just leaves the final mass of the bioweapon to consider. A critter weighing in at 300kg would be frightening, but is not necessarily all that practical: it takes longer and more nutrients to grow, but is a bigger target that is more easily killed or spotted and avoided. A bioengineered creature weighing around 7.5kg could quite easily be ten times stronger than a 75kg human, given that its musculature could also be bioengineered. Given an exoskeleton and a robust arrangement of internal organs, such a creature, perhaps the size of a small dog or monkey, could be expected to easily overpower and kill most humans all by itself... and such a small body size reduces the time necessary to grow it.
So, to go from 128ng to 7.5kg would take around 36 doublings, at 40 minutes per doubling, for a total time of 24 hours. Add the 35 minutes for initial cell division to 128 cells, and you have your dog-sized horror in just over one day.
The problem is going to be providing it with energy, nutrients and dealing with the waste heat from all this furious cell division. Supplying nutrients isn't an insurmountable problem: growing within a host could easily provide the necessary nutrients, and it's not as if the host has to survive. However, dealing with waste heat could be a problem. A live human host is actually ideal, since humans have a practically unmatched capacity for shedding excess body heat amongst all large land-dwelling species. However, given the rapid growth of a 7.5kg parasite, the death of the host is quite likely, and the corpse would become insulation rather than a cooling mechanism.
So... it may be necessary to reduce the growth rate of this bioweapon just so that its growth does not cause it to overheat. This is an actual problem in large commercial fermentation tanks: they must be actively cooled so that the rate of cell growth and division does not cause overheating and cell death. However, a reproductive cycle on the order of 25 hours to perhaps a week at most is going to give it rather short generations. Let's say that it has a growth time of 5 days.
By taking an r-strategy over a K-strategy, with short generations and a large number of offspring, such a bioweapon might be designed to implant its embryos into a large number of hosts. If each individual implants ten hosts and each host produces two offspring, and we start with one individual bioweapon, in 5 days, we have 20, in 10 days there could be 400, in 15 days 8,000, 160,000 in 20 days, 3,200,000 in 25 days, 64,000,000 in 30 days, 1.28 billion in 35 days, and 25 billion in 40 days... but if we're using humans as hosts, humanity would be extinct before then.