I'm not aware of anything in your overall price range ($50), but you should look at Touch Control
. While the App is well within your price range, you'll need to replace your MRF-350's and you may need to purchase iPad/iPhone(s).
A system of your scale is not a problem. You would replace the MRF-350's with Global Caché units that are WiFi or wired and can be scattered about as needed. Need to sense when the garage door is open or send IR to a device that is difficult to wire back to a rack? -- no problem.
My programming style for any control system is to allow any remote to control any room. You can then use one remote per area or one remote per user. If the dog eats a remote, you can still control the whole system with one of the remaining remotes.