So I have been tasked with getting alliances for Bruma, but I was also tasked by the emperor to retrieve the 4 things to open a portal to 'Paradise'

I have gotten the 3 things and now I need to open the great oblivion gate for the sigil stone which is the 4th thing.

Is it a bad idea to get the fourth thing before getting alliances? Or should I get alliances for Bruma first

I am almost certain that you cannot get them in the alliance without closibg it, but you have to ask them about joining the alliance for them to send their guards in the great gate with you.

Its either the above or, get them to join first then an army arises to join you in raiding the oblivion gate.