You can file and fly in the TMA with single engine , just be prepared to have a load of altitude to get rid of . They will drop you out right above EGTR unless you request an early descent , then they will route you back out towards BPK or LAM.
Clarification on the GPS , it has to be an approved procedure and GPS cannot be a portable unit , it must be certified for approaches . At LYDD for example the MDA is 430 on the ILS and I think its 460 or something like that on the GPS21 Y . That means should you need to return for some reason you will not have the MDA before breaking clear of cloud .
LAM works quite well into EGTR and I would not go below +- 800ft as said before , there are a few tall buildings at 2 miles for 26