Almost all 9mm ammo is fired from semi-auto handguns so I would think it would certainly be safe. The only question I would have would be if the 77 grain payload would be enough recoil to operate the action.
During the late 1970s our issued pistol was the 39/39-2 and issued ammo was the Federal 95 gr JSP. We clocked that round out of our guns at 1400 fps. We never had any problems with that round functioning in our guns.
Reading Doubletap's website they list their 77 gr at 1600 fps out of a Glock 17.
I'm not a fan of going that light of a bullet or some of the small boutique companies out there but you can only shoot what you can find. I could not find the shape of their 77 gr bullet. If it's round nose or nothing too flat I suspect the Doubletap will function fine as long as it will feed. Some guns tended to be picky on feeding short OAL.