If your cages are emitting a lot of odor perhaps you should consider switching bedding, or consider spot-cleaning after they relieve themselves (a black light flashlight can help you see where they urinate). I stay on top of that religiously and the cage never smells.
However regarding a purifier- I would seriously recommend constructing your own rather than buying one.
I use a 6" carbon filter with a variable speed 6" inline fan. I also built a filter box to hold furnace filters that sits on top of the fan so the carbon filter does not get clogged up with dust. It is very simple- just a cardboard box with 12"x12" furnace filters taped to it, sealed air tight, and a 6" hole cut in the other side, so it sits snugly over the fan. I have one cheap pre-prefilter furnace filter (MPR 300) and one expensive top-of-the-line furnace filter (MPR 1500 or higher). I considered going full HEPA for the second prefilter but decided this was sufficient.
https://www.amazon.com/VIVOHOME-Inline-Control-Australia-Charcoal/dp/B07XD75T6H
Depending on the size of the room a 4" may suffice
https://www.amazon.com/Filtrete-Healthy-Allergen-Reduction-Furnace/dp/B005GZ90QY/
This is the 1500 level- there are better ones available but this should be enough. They also sell these singly at home depot or even walmart sometimes.
Then you can get whatever the cheapest thing is you can find for the first prefilter.
Some people may run the carbon filter by itself but that's a mistake because it will quickly clog up with dust and shorten its lifespan considerably.
The way it should look, with the air moving from left to right is: Cheap filter > Expensive filter > Fan > Carbon Filter
If you decide to go with a prebuilt "air purifier", you will probably waste a lot of money on filters over the life of the unit. The small carbon filter models do not last long. One thing about furnace filters- they pretty much cost the same regardless of what size you buy. If you have the room to make your prefilter box larger, you can use larger filters that require less frequent changes. A 12x24 costs nearly the same as a 12x12. And by having two prefilters, you extend the life of the expensive one by changing the cheap one out first and more frequently.
The only downside to this is the noise level- but getting a variable speed fan will allow you to have it on low speed which helps a lot.