A bit late of an answer, but it is definitely feasible to replace the main board, though it does require a decent bit of disassembly to be able to access the main board. You'd have to remove the back, front, and top covers to be able to get at it. This guide will take you through all of the steps necessary to replace the board.
As to finding a replacement, you can just search up "Canon Rebel T3 main PCB board" to find sites that sell them. eBay is one place that you'll likely find it being sold at, and probably one of the cheaper places you can get one from since you can find used boards being sold there. You can also try searching for it using its part number, which is "CG2-2977".
As for the price, at the time of writing this answer it seems like you can find places that sell them for around $100-$150. I also did find a used board being sold on eBay for $65
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