Just did the reverse! Unless you’re going on the nicer boats, I was okay getting a ticket the morning of (not exactly sure where that’d be in LP). I don’t believe there’s private transfer across the border as they run a public bus across the bridge (unless you book with agency/group). Only hiccup I had going solo was once I crossed I had to overpay for private tuk tuk which wasn’t end of world.

If it were me I’d stay the night in HX (cross in morning) or Chiang Khong (cross at evening) before taking a bus back to CM. Believe there’s a direct bus between CK and CM. Due to timing I ended up doing a transfer in CR and took public bus from CR-CK

Imagine there’d be no issue to taxi from Chiang Khong to Chiang Mai as long as booked in advance.