Getting Married while appealing EEA2 UP RC Refusal?

My girlfriend (legally working Bulgarian exercising treaty rights in UK) and I (American National) have been in a relationship and living together since May 2012 (not quite 2 years). My EEA2 unmarried partner residence card application was refused on 3 March 2014 solely because of the supposed 2 year minimum co-habitation requirement (this was specifically mentioned in the refusal letter).

I now have to appeal this decision or be forced to leave the UK and my girlfriend as I came here as a tourist in July 2013 and those 6 months were up in January 2014 (we were living together in the US before I came here with her). I have hope that if my case came before a judge who actually looked at all of our evidence that they might reverse this decision as EEA regulations do not specify such a 2 year minimum requirement.

but.....

My questions are:

Can my girlfriend and I just get married after I lodge the appeal, cancel the appeal once married, and then apply for a resident card on the basis of being a married partner?

Am I correct in assuming that once married, applying for a resident card as a married partner would be much faster (only 2 - 3 months) and more guaranteed than waiting on the appeal (8+months)?

We have been considering marriage anyways and do not want to have to wait another 8+ months for the uncertain appeal if we don't have to.

Any insight or thoughts would be greatly appreciated!

Getting married is your natural right.
I thing that's a good idea to strong your case rather then long wait.
After marriage applying RC will be faster route.

Thank you for the reassurance!

If we do indeed go the marriage route, will we run into any issues regarding them thinking it might be a marriage of conveinence?

Are they going to ask about my refusal / appeal and will them knowing about my situation affect their decision to let us marry?