If the bag is that small, and causes your hand to get swaety, the possible sweat droplets should be your main concern.
If the film you used in not rated above ASA1000 or IR, you probably could forget the changing bag entirely, and use multiple sheets of dark fabric/blanket, in a darkened room, with the lights in the adjoining room all shut off just to be safe. Use the blanket as the layer closest to the camera as the lint of most blankets often help to break up any possible scattered light rays (if any managed to get in in the first place).
In my early days of a poor-man's 'dark-room', I got away with much less light-tight security. Also, use your arms, body or whatever to bear down on the surrounding edges of the material as a further safety measure.
If you're still not confident of doing it, a possible option is bringing the unopened camera to the lab, tell them what happened and ask them to extract the film in a changing bag...but unless you have a very careful technician, don't expect them to apply extreme caution or delicacy handling your film....or camera. Chances are, they will not be responsible for any inadvertent damage either to film or cam.
All the best...and thanks, in the past for sharing with me your focusing screen modification instructions for DSLRs.
CHEERS!