How can I fix a relationship with a professor that has recently cooled?
My general impression is that professors quite like giving advice (it's almost a job requirement...) but are incredibly busy. I would thus suggest you do not try to change 'your relationship' but focus on the task at hand, i.e. directly but politely ask whether your professor would be available to give you some advice.
Of course there is the possibility that your relationship is so bad that your professor will not even consider meeting with you. In that case I would not try to change his/her mind. Just treat it as a lost cause and move on.
In the more likely scenario of you managing to schedule a meeting it's important you are prepared and respectful of his/her time clearly signaling to your professor that his/her advice will be not be wasted. If you are clueless about your questions that will quickly signal to your professor that s/he is wasting her time. If however you seem genuinely interested and it looks like you are indeed benefiting from the advice I wouldn't be surprised if your relationship improves significantly and could even develop into something more formal like that of a co-advisor.
Either way: simply ask that you would like some advice, ideally explaining what makes his/her advice valuable to you. You'll find out how to proceed after that. Good luck!
Request to meet with the professor. Make it very clear in the email/phone call/carrier pigeon note that you are looking to apologize for previous poor behavior/performance and want to make amends. He is busy, and he does not sound like the type to suffer fools; your emailing him is already making him suffer this fool (you) again, you need to clearly make him understand that you are apologizing.
When you meet, apologize. A good apology has three parts: (1) I'm sorry I screwed up. (2) I recognize that the screwup was my fault. (3) I won't do it again. You should offer all three.
If there was a good reason for poor past performance (personal illness, family situation, etc.), you can try to explain past behavior, but ONLY in the context of "...but that's no longer an issue so now I can be better." If you can't state that it won't be an issue don't bring it up. However, you will have to offer some form of "and here's why I'll be better this time around," because otherwise he has no real reason to take you on again.
Discuss your current request and ask whether he would be willing to help. Don't be upset if he says no or tells you that "he'll get back to you", which means no. Be polite and thank him for his time.
I would advise you to just knock that professor's office door when you feel the need to meet him. Sending him an email to ask for an appointment could give him the opportunity to delay the appointment or to just answer you he is very busy for the next two to six months. When I feel the need to meet a professor of my department I just go and knock his office door. Generally, professors will listen me first, and if they are not available at the moment to talk to me, they give me an appointment. That is, I believe the best way to proceed.
I believe it will be scary for a professor to consent being a regular adviser to you while not being your official advisor. He may be scary to accept the title of regular consultant or advisor you want to honor him, thinking that your official advisor could apprehend that as a challenge or a competition to to him.