If the kitchen sink is a distance away from the gray tank, it isn't unreasonable for that line to have frozen under the trailer somewhere between the sink and tank. Warming under the trailer to more than 32 and wait or try to drain the line and flow hot water directly to the plug with a tube or hose are options to thaw it. Also, there is a possibility that the kitchen drains to the black tank and could be part of that whole issue.
Since the toilet worked during colder temperatures but is now plugged when relatively warm, that just points even more toward a poop plug in the tank.
After reading about others struggling plugged tanks, I came up this crazy plan just in case it was needed one day.
1. Close the black tank drain and disconnect the drain hose. Make sure the hose is empty first. If not, the plug is down stream from TT and would have its own remedy plan.
2. I'd connect one of these with the gate valve closed.
Ready to Use Gate Valve with Built-In Clear Adapter
3. Then attach one of these.
RV Sewer Cap with hose connection
4. Then use a section of garden hose screwed onto the cap that is long enough to reach the septic inlet with the open end higher than the toilet. But, it has to be shorter than the drain snake in the next step.
5. Then run one of these down the garden hose to start poking holes in the poop pyramid.
6. Open the gate valves and go to work. Once there are fluid returns in the clear section, pull back the snake and close the drain gate valves. Use the garden hose to drain the liquids down stream of the clear section. Put the drain line back on and see if it dumps.
7. Repeat until it dumps or a good enough hole is drilled in the plug that water can be added to the toilet to begin eroding the poop lasagne. Adding warm water with dawn dish soap and letting it soak has been reported to help rehydrate and dissolve mount pooparoonie.
To minimize collateral damage, I'd spring for a tyvek painters/hazmat suit and good waterproof gloves as this has the potential to get shi**y in a hurry. If you try it, I'd love to hear the war stories.