Don't do it
You are the wrong side of legal -
If you want to claim the hours on a group-owned aircraft, where you are not a member, then pay up.
It doesn't matter that the group's paperwork is 'unofficial'. It will prove to be evidence in any further claim.
Just remember that if anything went wrong and the CAA check your logbook and that of your friend and the group's records -and they would - the insurance might be ruled invalid.
What absolute B*ll*cks, the insurance is a matter of you being approved by the insurance company which the group can sort out - paperwork on a group aircraft has no standing unless it is a way of "signing out" the aircraft because even that does not necessarily designate the pilot in command (unless of course that is designated on the paperwork - and group rules can allow for non members to be P1 on the paperwork), the only real way that you would be in trouble, provided everything was done as suggested, with approval of the group and correct shares of payment, is if both pilots logged the time - which is ruled out in the initial post - the main problem that I see is that as said already is that there can only be one Captain - and if you are logging it, then legally you are the Captain, the way it is being done at present then the group member is Captain and this would legally have to change, no problem in doing this as long as he and the group are happy with it.In practice a gentlemans agreement might be that the group member is actually the boss, but this would have no standing in law.