I've stayed at most, but not all, of the downtown hotels and can make a few comments.
1. In general, I don't think your budget will be a problem, but as you might expect rates can vary pretty widely. Thus, the specific dates you're talking about can be semi-important. If there is some major event going on, you might have a problem meeting your budget.
2. Just to be certain you're aware, you'll have to pay for parking at pretty much any of the hotels. It's not as step as some major cities, but as my memory serves me you're talking ~$25-30/day and a probably a little more for valet, but that will also give you in/out privileges. Really not anyway to avoid it (unless you go with the La Quinta, but I don't recommend it, particularly because I don't think they have a pool). If your budget is supposed to include parking (and tax, see next item), you need to factor that into your planning.
3. Don't remember what the total tax on hotel rooms in Indy is off the top of my head, but in some cities in can be in the 15% range. No real way to avoid it, but you'll probably want to undershoot your budgeted figure when looking at rates if you're trying to get the after tax total ~$160 (and if you factor in parking . . .).
4. Regarding the pool, just be to certain (again, we don't have your dates), you're probably looking at an indoor pool. I don't believe any of the downtown hotels have outdoor pools. Also, I'd be sure and check photos on the pools as they can vary widely. For example the Omni has a pool, but as my memory serves me it is pretty much a postage stamp sized affair and might even be in the basement.
5. I know the Crowne Plaza has a pool that is more reasonable, both in size & location (essentially in the lobby, but not at the front). The other kind of neat thing about this hotel is that it is in the old Union Station and has actual restored Pullman cars as some of its suites. The downside is that the "windows" of the rooms are simply to the lobby area and some people complain about the train noise (as it's next to still working rail lines).
6. I'm sure the Marriott has a pool, but don't have a specific memory of its layout, size, etc. I'm thinking it's in some kind of atrium, but aren't 100% sure.
7. I've also stayed a few others (Westin, La Quinta and Sheraton(? but I think it was a Ramada at the time or maybe vice-versa), don't think I ever checked out the pools at these and/or they might not have one.
7. About the only other factor I can think of for you to consider is whether you'd want to walk to the zoo or not. If you think you might, then you might want to focus on the "Marriott row" located just north of the Convention Center. These are the closest to the zoo and are a reasonable walk (obviously those on the west end are close than those on the east end).
8. Not sure if they have pools or not, but there are a few hotels that are next to Lucas Oil Stadium. I'd probably avoid them, not that there is anything wrong with them, but they're *slightly* farther from most of the downtown stuff and even farther from a psychological standpoint as you have to go under a set of *very* wide railroad tracks to get to the main downtown area.
Beyond that, I'd say get on your travel site of choice and filter what they come up with based on needing a pool (indoors if necessary?) and see what the rates are looking like on your dates. As commented I would check photos (either on TA or the hotel's website) to see the actual pools to make sure it will be acceptable. I'd look over the TA ratings & reviews for your short list of hotel and check around to find the best rate you can. Also, you might want to call the hotel directly to be 100% certain the pool isn't under repair or something. (I'll close this part by pointing out that in my opinion, unlike many other cites, there aren't any really bad hotels in the immediate downtown area.)
Lastly, Hotwire is a possibility and I regularly get a reasonable deal on downtown Indy hotels, but I'm not sure I'd use it in your situation given that the pool is such a priority. While you should be able to pick a hotel with a pool, you might get one that has a pool "in name only" and/or that is under repair.
Hope this helps,