What is the ruling on going to a gym where there is loud music, while committing not to listen to it, in the absence of other alternatives?

This music is one of the reprehensible acts that must be denounced. If you are able to denounce it to the officials in a way that removes the evil, then it is permissible for you to go. Otherwise, do not approach the place. Denouncing with the heart includes leaving the place of evil if you cannot change it, in accordance with the verse: "And He has already revealed to you in the Book that when you hear the verses of Allah being denied and ridiculed, then do not sit with them until they engage in another conversation. Indeed, in that case, you would be like them. Indeed, Allah will gather the hypocrites and disbelievers in Hell all together." And your need to lose weight is not a preponderant benefit that justifies going, as there are many alternatives for weight loss without any forbidden religious implications.