Well you are between a rock & a hard place here ;-{
At this point I don't think it's worth swapping out if your system is working. You can buy a used system for less than the cost of a replacement logic board (approx $750 U.S.)
As to your direct question you can replace the logic board within the same series without issue. In this case the i5 logic board (Macmini6,1) can be swapped out with either i7 (Macmini6,2) models being an exception to the general rule.