Can weight machine training or certain exercises cause more damage in vaginal prolapse after menopause? Will a Pessary or tampon help during exercise?

Weighty problem. Yes, the weight machine will cause you to hold your breath and further push down the vaginal prolapse. A pessary inserted may work or not, but the tampon will get pushed out by the exertion and is of no use whatsoever to stop the descent of the vagina.