As many have pointed out so far, Mega Maid has some physical limitations. It's almost like it's just a plot point in a movie or something! But that doesn't mean there's not options.
What you are looking to do is similar to what we have to do in real life in clean rooms. In fact, it's something similar to what we do with normal air conditioners as well:
A critical factor in cleanroom design is controlling air-change per hour (ACH), also known as the air-change rate, or ACR. This refers to the number of times each hour that filtered outside air replaces the existing volume in a building or chamber. In a normal home, an air-conditioner changes room air 0.5 to 2 times per hour. In a cleanroom, depending on classification and usage, air change occurs anywhere from 10 to more than 600 times an hour.
10 to 600 air changes in an hour is a lot! That's way more than your half-an-atmosphere-in-a-week criteria, though admittedly for a much much much smaller volume of air. But they also have an interesting criteria called "laminar flow." Basically, the air circulators in a clean room aren't allowed to create any turbulence. If they created turbulence, the eddies could suspend particles for an unknown amount of time, but if the flow is smooth, the lifespan of a particle in the air is well defined, and quite short.
You're not going to need laminar flow to do your job, but your calls to not change the weather are very reminiscent of that requirement. Weather typically goes hand in hand with chaotic turbulent flow, so if you minimize the amount of turbulence you create, you're going to minimize your weather issues. You can take lessons from the clean rooms and apply them to save your dinosaurs!
One of the big keys to how clean rooms do their job is distributed air handling. They don't have just one air intake on one side of the room. The area is peppered with holes for the air handlers to pull air out of the room. The ceilings also are designed to maximally distribute the air across many openings so that there is an even flow of air from ceiling to floor, pulling all the dust out they can.
You should do the same. Now I don't know what your budget is, but given that you've gone back in time, I'm assuming you can predict the result of enough football games to cover some pretty outrageous bills. This should be enough to build enough hardware to distribute your megamaid across the entire earth.
Now don't get me wrong, you're going to do some environmental damage. Let's say we want out air handler to take up 1% of the total surface area of the earth. That's going to involve cutting down a lot of forests, but it's distributed so you should be able to do so without causing the extinction of too many species. Install the the "return vents" near the ground this way, and we'll put the outlets way up high. The stratosphere should do the trick.
Now our air intake area is pretty massive. It's now 5,100,000 square kilometers! As Chris M. pointed out, the atmosphere is 4,200,000,000 cubic kilometers, so to process that in 7 days you need to process about 7000km^3/s. Now seven thousand cubic kilometers of air is a lot of air to move in a second, but we get to move it with a substantial distributed air intake, so the air flow rate only has to be 1.3 m/s!
Now that's a noticeable 3mph breeze, straight down, so it will definitely have some effect on the weather, but a manageable one. If you can expand your budget by predicting a few more UFC tournaments, and can take up 10% of the planet's surface area, you can bring it down to 0.3mph, which would be unnoticeable.