The lesser known superviosr has a grant for you and it is from the EPSRC, am I right?
In that case, here's my input for you, if it is not too late or else for anybody else.
The EPSRC grants are very competitive, not for the student but for the professor to get hold of. First if only he has a decent publication record and popular among other scientists, he could submit a good proposal to the EPSRC with an innovative idea.The EPSRC then would send it for peer reviewing to three independent senior professors in the field. Unless the reviewers agree almost unanimously,the proposal won't be a success. So in this case, the lesser known professor should have done a good job and the Oxford professor might well have refereed his proposal and approved it.It'd be my pick.

The inexperienced supervisor is likely to give you their all unlike the well known one... it’s almost always the case.
Funding competition is stiff - and if they have funding it simply means they had a good proposal plus with funding comes greater accountability - they will make it their business to see that you deliver because if you don't it will reflect very badly on the department. Plus bear in mind that an experienced supervisor usually co supervises with someone with experience... I doubt they will be alone on it with you. The funded place would be my pick - I was self funded for a whole year before funding came through and it was TOUGH.
Good luck making a sound judgement

In case this is still relevant. I'd go with the one that makes you go oooooooohhh, because this is the foundation for your career and what you'll be focussing on throughout that. Also Oxford will open, not close doors for you. I did my MA at an internationally top of the league uni and it certainly did that for me. People see you in a different light, wrong or right that is the way it is, and that will last forever. And I'm sure it will be worth the money. Just my POV mind.