A Center Controller reported a non RVSM military aircraft flight was approved to climb to 30;000 feet underneath an Air Carrier at 31;000 feet.

Narrative

I was working DSide 88/97 on the 97 scope. The low sector; 98 was split off due to traffic saturation. Thunderstorms had kept the 97 and 88 sector busy all day. 98 Rside attempted to call over the line but ended up hanging up and verbally coordinating an unrestricted climb into ZZZ1 up to 41000 ft. 97 Rside told 98 Rside about traffic at 31000 ft. and 35000 ft. 98 Rside acknowledged traffic. Shortly after; the ZZZ1 airspace was turned on. Shortly after that the Aircraft X flight became airborne and we got the automated tag. Under the assumption Aircraft X was only climbing to 29000 ft. because it was Non-RVSM; 97 Rside accepted a point out of Aircraft X and flashed the data block back. We both observed Aircraft X flight's data block climbing quickly. At least 5000 FPM. Again; we assumed Aircraft X would be stopping at 29000 ft. so Rside did not call traffic for Aircraft Y at 31000 ft. Aircraft X data block jumped from 290 to 299 and simultaneously Aircraft Y reported it was responding to an RA due to traffic less than 100 ft. beneath them off of their left side. Aircraft X flight was not inside the ZZZ1 airspace yet so the Non RVSM was absolutely a factor. 98 Rside admitted to releasing Aircraft X unrestricted to FL300 and did not realize the Non RVSM. I'm not sure what altitude the Aircraft X stopped its ascent. I think most of the time this isn't an issue. I believe it was mainly 98 Rside not realizing or overlooking the non RVSM of Aircraft X. If there were any procedures to be changed it would be to only allow unrestricted climbs to the top of each sector and coordinate once the aircraft is airborne. However; again; I think the climbs into the ZZZ1 areas are usually fine and this particular incident was on the Controller.