My "low-end" older Denon 2106 can be configured so it's surround amplifiers are used for zone-2 power. You can also select a different source for zone-2. In this configuration, the sub would only work on the primary zone, so you'd connect the sub to the sub-out, the kitchen/living room speakers to the front left/right speaker outs, and the other two speakers to the surround amplifer. Then set the surround amplifer to zone-2 and you'll have power to all your speakers from the one receiver.
I took the liberty of perusing the Denon AVR-1610 manual and, though Denon's site says it has "Enhanced Power Amp Assignment" I don't see it in the manual. On my Denon receiver the option is "Power Amp Assign", I'm not going to look through all their manuals for their newer receivers to find the least expensive model that has it...I'll leave it to you, but I'm going to doubt you'll find a receiver with the option for $200.
Your other option is to use a single stereo amplifier (which will obviously force all the speakers to play the same source). Then the volume controls will balance the output between the pairs. If you're OK with having every speaker pair play the same thing, this will probably be within your budget and any reasonable stereo receiver will do.
Edit: HOWEVER, a stereo receiver may/may not have sub-outs and even if it did, you'd be depriving the zone without the sub of the sub signal by engaging the x-over. So...you'd want to find a stereo receiver with pre-outs and use them for your sub and use the sub's x-over. But, you be doubling the bass (the speakers in that room would also get the sub frequencies). Alternatively, if the stereo receiver had an "enhanced bass" option (which basically means put the bass to both the speakers and the sub) then you could use that too, but you're still doubling the bass.... Essentially, unless you don't mind either a) losing the sound below your x-over point in the room without the sub or b) doubling up on the bass in the room with the sub, a stereo receiver + sub option isn't really going to be easily implementable.