So how do modern trumpeters deal with low Cs in Mozart's and Michael Haydn's music?

The trumpet in the Classical era was a valveless instrument. This meant they could only play notes from the harmonic series. The trumpet in C which would have been used in Mozart's 'Jupiter' Symphony would have been eight feet long and have had a pedal tone of C two octave below middle C (otherwise those low G's in the second trumpet part would have been outside the range). Therefore, the C below middle C (as in the minuet of the Mozart symphony) would have been the very playable first harmonic on such a trumpet. I can only assume the C two octaves below occurs in the Michael Haydn symphony (because it doesn't in the Mozart) and still puzzles me. The pedal tone of a trumpet is virtually unusable due to its difficulty to play by the musician (due to the small mouthpiece) and poor tone quality, so I can't imagine why Haydn would have used it.

As to what a modern trumpeter would play, their only option is to play the notes an octave higher.

The C below middle C is entirely possible on a modern trumpet, so long as the player is skilled enough. Pedal tones are not used frequently, as they are of poor sound quality, and the timbre is disgusting (I liken it to a fart)... not to mention the difficulty of accurately producing that pitch. so most people play up the octave.

I do not know what Haydn piece you are referring to with the C two octaves below middle C. Care to clarify?

Siegmund, could you give the bar reference(s) in K551 in a supplementary, so we can see what you are referring to?

All the best,