I suspect that taking advantage of the BA holiday double tier points on a number of sale fares could be the cheapest way. The BA holiday scenario imagines that you aren't starting from scratch - otherwise getting to Gold would need a number of resets. The double points offer is also only open via the UK.

A DONE3/4 would be probably GBP3k / USD4500 including taxes even if super optimised.

BA flights to OTP (Bucharest) from London are probably only about GBP500 for 80x2x2 TPs (320TPs/flight), so three of these flights gets you BA Gold (if you're already Bronze/Silver), noting that BA needs a minimum of four BA segments to move up tiers.