A piece, broken down into parts, typically consists of a melody and accompanying chords. The puzzle then becomes a bit more well-defined - how to lay the melody on the strings so that it can be played fluently while having some* access to the chords as well.
A melody on the lap can move in three dimensions. Along a string, continuous in pitch and sound (if needed), done by moving the bar. Across strings, which makes a non-continuous note change but can allow multiple notes to ring out at the same time and does not require moving the bar. And then there's the hybrid approach where rotating the bar can allow a mix of both. I prefer to leave bends out of scope.
From those approaches the easiest one to play fluently and quickly is probably across strings - no bar movement pause or a unintended glissando and no risk of moving the bar to the wrong place. Moving the bar across the strings comes at solid second place as it incorporates either a pause while the bar is moving or a pitch shift that may not be suitable. Rotating the bar is quick but has its own challenge of doing it well enough to be in tune.
Here come's the first conflict of the puzzle. We typically cannot play an entire melody while on the same fret, much less accompany it unless we are named Bill Hatcher, play on a 14-string lap and use dual bar technique
We have to move the bar and the melody and the tuning will dictate where we can move it to. A quick melody, in my opinion, has to happen across strings with pick blocking and little bar movement. A slower one with longer notes can tolerate pauses and glisses while one moves the bar back and forth.What I did for the tuning of my choice was to create two kinds of diagrams for it. The first one is the typical fretboard chart of all the notes. The other ones showed scale degrees on each string as if it was the first degree of the scale. E.g. for the 6-string C6 that'd look like
(1, 3, 5, 6, 1, 3) (b6, 1, b3, 4, b6, 1) (4, 6, 1, 2, 4, 6) (b3, 5, b7, 1, b3, 5) (1, 3, 5, 6, 1, 3) (b6, 1, b3, 4, b6, 1). This helps to figure out the chords grips for any root on a given string. Armed with that the puzzle looks a bit more approachable. I pick a starting fret based on having maximum notes matching the melody and from there on I try to keep it vertical. If at any time the chords don't work or the melody allows it I switch to laying the melody horizontally. As for slants - I'd use those only to get chords not usually available and try to avoid them for melody notes.
On chords, those will need to be drastically simplified. A root or the third/seventh of it will make a diad with the melody note, I'll try a few options and leave the best sounding ones. Three and four note chords will usually be standalone, not really participating in a melody.
I know that are many professional musicians on the forums here that have arranged multiple pieces for various tunings. If anyone feels sharing there thoughts about it please do. Cheers!