Look around, there are deals to be had.
If your trees are truly 12" and smaller, you ought to go buy an Echo CS346, which you can find NIB on eBay for less than your budget price. Pull out the muffler screen and retune, the saw will get its oomph back that way. With a loop of non-safety chain you will have a durable, reliable little machine that is ideal for the homeowner and is actually a solid performer.
Here, in top-handle trim:
There are other options, too, in the way of older saws. I just bought a Husqvarna 154 that runs quite well, for less than your budget amount. It would be way more saw than you need, but is a durable, proven design capable of helping out friends and neighbors with bigger projects, if the need arises.
And also ask around at the local saw shops, OPE dealers, and the like, you might be able to find a killer deal on a NIB New-Old-Stock saw, too. I scored this little guy earlier this year, again for less than your budget amount. It had never so much as seen fuel, and still had its "safe use instructions" tag hanging on the handle. Killer little saw with this 13" Narrow-Kerf bar, great for limbing and smaller trees!