As an FYI from someone who has been paying tuition in GBP for the last 2+ years, Amex seems to be hit and miss when it comes to foreign transactions counting towards MSR. I have had one success and two fails. The first one almost cost me a SUB, but I caught it with 2 weeks to spare so I could make up the spend. I argued with several amex reps over this, but none could really explain why the spend was not counted for MSR purposes, though one rep speculated that somehow the currency conversion process may make it look like a finance charge. Like I said, I gave up and met the spend elsewhere so I didn't pursue my argument to its conclusion.
I have had no issues with Chase on GBP payments. I am currently working on the Cap1 VenX MSR and have included GBP tuition payments in my spend, but i still have $4k to go so I don't yet know if there will be any problems. Cap 1 VenX could be a good choice for you as they seem to give good credit limits - $30k seems fairly common.
As for Chase BA, could work for you particularly if you have use for the 'Travel Together' voucher you get after $30k spend, I think. But taxes on BA awards can be really high for long haul. I tend to use my stash of BA Avios for short haul flights in Europe.