Anything is worth a try, but in my prior long experience of immigration stuff, emergency passports are really issued by states for the purposes of single journey repatriation only (with possible transit points en route - which may not permit exit from the transit airport eg Piarco).
So be aware that the Guyanese may issue him an emergency document with, say, 30 days validity - but it may not be accepted for boarding by the airline to T&T. Or if it is, he may be refused leave to enter at Piarco - and the Trinidad immigration detention centre will not offer the kind of comfort or vacation pleasure he is looking for
All that said, this is the developing world and I am 10 years out of working in the field, plus it is possible that Guyana and T&T have some kind of reciprocal arrangement so it is definitely worth going to see both the Guyanese and T&T consulates. I will be interested to hear the result.
NB: his final option will be to ask how long the Guyanese passport department takes to issue new passports in Georgetown itself - if it can be done very quickly back at the home country main office, Plan B would be for him to get an emergency 1 way ETD to get him there where he can get a new full passport then zip over to T&T for the vacation