I assume you have checked if the game responds to a controller in port 2 or something, quite a few will.
Hopefully you thread on replacement ports comes to something. That said shutting off sounds more like short protection, what is wrong with the port? Also are you sure it is the port and not the pad?
Anyway yes it should be possible to hack the Wii, there are four things I would consider
1) Manually rewire port 4 into port 1's points. Granted this is not a software fix but I have to mention it. There are several ways you could set about this, looking at (
http://tr1.cbsistatic.com/hub/i/2006/12/07/4f3061c9-c3bf-11e2-bc00-02911874f8c8/38943.jpg , the best underside PCB shot I could find) it is fairly basic soldering.
2) I am not sure how far the use a GC pad instead of the wiimote stuff came along as far as the general public being able to buy/make something. Again not a software thing, this would come with the bonus of being able to use the GC pad even on games that did not support it.
3) As you are on the Wii a lot of things are governed by IOS (wii) and MIOS (GC stuff). This would require big boy hacks (one does not edit IOS or MIOS without knowing what one is doing) but if you are actually using hacked games (or can be persuaded to) then this could be good.
4) Hack the games themselves, 3) would do for all games and might even be able to do some 2) like stuff if you are really good but let us not go there right now. Anyway this would need to be done on a per game basis. Reading chapter 9 of
http://hitmen.c02.at/files/yagcd/yagcd/frames.html and
http://wiibrew.org/wiki/Hardware/Serial_Interface it would seem it is not likely to be that hard and may be able to be done with cheats (all the reads, protocol and decoding stays the same, you just have to make sure it reads from port 4 rather than port 1*).
*note that the Wiibrew docs start numbering at 0 so you do actually want to read at 3 rather than 4, the maths stays the same though.
All these require some effort though, I am not away of any simple things you can do as far as homebrew launchers or StartPatch/starfall type options.