Does melatonin supplementation suppress endogenous melatonin production?

Nope.

References

1. Matsumoto M, Sack RL, Blood ML, Lewy AJ. The amplitude of endogenous melatonin production is not affected by melatonin treatment in humans. J Pineal Res. 1997 Jan;22(1):42-4. doi: 10.1111/j.1600-079x.1997.tb00301.x. PMID: 9062869.