Braham-Barrett qualifies through his late paternal grandmother and he was requested to represent the Caribbean island for World Cup qualifiers against Curacao, away on March 27 and at home on March 31.
It would have meant missing Cheltenham's home SkyBet League Two match against Plymouth Argyle on Saturday, March 28 and Braham-Barrett, who could also represent Jamaica through his grandparents, wants to stay and fight for the cause.
"I'd love to go to the Caribbean and play international football, but the situation we are in means it is not an ideal situation," Braham-Barrett said.
"I have thanked them for their invitation, but told them at the moment I want to keep playing for Cheltenham and not lose my place in the side.
"The squad is looking much stronger now and I want to be a part of it so it would be unwise to walk away and miss a game."
Braham-Barrett first heard of Montserrat's interest at the end of February and he said: "I am proud to even be considered, but now it not the right time.
"I have spoken to the coach (Lenny Hewlett) and he understands why I want to stay and help Cheltenham climb out of trouble.
"It would be selfish to go and he's told me the door will still be open for me in the future.
"I know Dean Morgan (on loan at Crawley from Woking), Darnell Furlong (QPR) and Antony Griffith (Carlisle), who are also in the squads, so hopefully I'll have a chance to link up with them soon."
* New loan signing Will Packwood will miss the Plymouth game due to his commitments with the USA Under-23 squad.