B737-700 Captain reported at cruise altitude the #1 engine began multiple consecutive compressor stalls. Engine thrust was reduced; ATC gave the pilots priority handling; a diversion was commenced; and an overweight landing was accomplished.

Narrative

During our climb to our cruise altitude we were give FL 280 due to an aircraft at FL 290. Normally our aircraft would level without any input from me; but we were climbing at a high rate and approaching the aircraft right above us. To avoid a TCAS warning; I attempted to shallow the climb; but the Autopilot was already in capture mode. I reduced the thrust to about 50% to help reduce the climb rate; once we leveled and applied power we got multiple consecutive compressor stalls from the #1 engine. I immediately recognized the stalls and reduced power until it stopped compressor stalling. Several times I tried to add power; but got the same multiple consecutive compressor stalls. While at FL 280; only the #2 engine was used to keep us at cruise. After accessing; we determined to divert to ZZZ1 and [requested priority handling]. Once we got to a lower altitude; the #1 engine operated normally and was not shut down. We did the checklist; I informed the flight attendants and passengers of the situation. We did a normal approach with both engines and an overweight landing. There was no brake cooling issues and taxied to the gate.