The stock cartridge on the Rega RP1 isn't the Bias 2, it's the Rega Carbon, the rebadged Audio Technica AT 91 if you like. If you get the Rega RP1 upgrade for about £85 then you get the Rega Bias cart which is a decent upgrade. In my experience, the Rega Carbon cart works well with the RP1: it certainly has more weight and authority than the Ortofon budget cart which was originally supplied as stock. Not to dismiss the Ortofon 0M 5E, which is certainly detailed if rather bright.
Therein lies the rub: the Pro-Ject Carbon tt is probably a better tt in many respects, but the stock cart is the Ortofon 2M Red which offers a very forward, some would say overly-bright, presentation of the music more akin, in my opinion only of course, to a cd. In my opnion again, the Rega Carbon has a better tonal balance more suited to vinyl reproduction. I will say here I have a 2M blue which is a very decent cart; I'm not an Ortofon-hater by any means.
Both are really very good budget turntables so what do you do? I'd buy the cheaper RP1 and replace the cart, eventually, with a much better cart like the Nagaoka MP-110 which would be ideal for the RB- 101 tonearm. I think that would make for a detailed, spacious and authoritative presentation of your records.
All my opinion of course based on listening experience with my ears in my rooms.