I go to an SSPX mass but don't consider myself aligned to any particular group, while I don't believe in BoD I cant speak for others since I don't ask them. If you are concerned with what the men at your chapel believe you have no choice but to ask them and see for yourself.
As for lifting weights, should a man go to a public gym to keep himself fit if that means putting himself in the occasion of sin? Because women at gyms dress extremely scandalously, no matter how strong the male's mind is he will eventually sin because the devil and concupiscence are stronger than him. Personally I won't risk my soul on this (and the saints say that only cowards are victorious over sins of impurity), and am waiting until I have my own house and home gym before I start taking my exercise seriously again, for now it's just very basic stuff.
Just something to keep in mind.
Gyms are one of the most unsanitary places, second to water parks. You don't have to have a gym membership to be fit, you don't even need an at-home gym. You can do little things to stay in shape, for example, when you're out in public, instead of taking an escalator or an elevator take the stairs, it's good for your heart and its cardio. You can buy dumbbells, it's a better and cheaper investment than spending money continuously on a membership. Go for an early morning jog (when you know no one is out), where there will be no women out to bother you.