Edit: I'll cover the edits in the question, but the below still is relevant to the question
Ignore the stuff about your peer. Your manager will not discuss your salary etc with your peer. If you are delivering things for the project, the PM will know and appreciate that. It sounds like there are maybe more issues than renumeration here though, it sounds like there may be friction between you and your peer and is that the real issue? I wouldn't worry if you decided to go elsewhere. If the job isn't a good fit for you then it happens. As long as you don't leave lots of jobs during probation then it isn't an issue.
The original answer below still stands though, it would be a tough sell to get a pay rise after probation period based on commute (when that should have been considered before accepting the job offer).
I can't personally see how you can justify asking for more based on the commute when you knew there would be a long commute when you took the job and agreed with the salary on offer. I assumed you signed a contract agreeing to the renumeration too.
The probationary period is a time where both employer and employee see if the role is a good fit. It's usually not enough time to then ask for a pay rise and isn't really designed for this in the industry (I get that call centres etc sometimes use this as a carrot to retain staff).
If you take the role permanently now you can then look at your Performance/Pay Review policies and follow those in 6-12 months (company dependant).
Could you cut down on the self training? Or use cheaper alternatives (i.e. Use PluralSight etc rather than face to face courses if you do this?).