You can leave London on Day 1, and leave Budapest on Day 2 to arrive in Bucharest on Day 3.

This would use 2 travel days. 

Example routes: https://www.seat61.com/Romania.htm#london-to-romania-via-paris-munich-budapest

It would then be cheapest I guess to buy a ticket from Bucharest to Constanta for cash.

For the return journey, you would use 3 travel days to get from Bucharest to London, so you could include travel from Constanta to Bucharest on Day 1.

I think a 5 day in 1 month pass would be best. (If you will come back within a month)