If you’re intending to redeem your miles with United then it might be better to use your Mileage Plus. That said, I’ll always join an airline’s FF program if I’m using them so as to hopefully avoid being bottom of the pile if things go wrong. That’s my thinking anyways.
If you do join KF, I don’t think that you will obtain any particular additional perks or benefits to add to your travel in J (I may be wrong in which case I’m sure I’ll be corrected

). And in my experience the KF mileage redemption rates are disgraceful (e.g. 2 return trips in J EU-Asia gets you IRO of 46,000 KF miles when 50,000 gets you Gold tier, redeeming 50,000 miles get you a $280 reduction on a $2,650 return ticket on the same route. Not worth it.
You mention that you have found a good deal or words to that effect. Before booking, you might just want to check the fare terms and conditions. There is not the same level of consumer protection with Asian airlines if your plans unavoidably change. Rescheduling your travel with Singapore Airlines is a chastening experience as I have found out to my cost. Don’t rely on the “Manage Booking” online, in my experience changing dates cannot be done online and you will spend hours on hold on the phone only to be connected and told “Computer says no” to any future dates you want to travel without parting with huge amounts of additional money in “Fare Difference” (in my case, double the original fare).
Bonne chance.
AF