What helps me a lot is thinking of the guitar/bari tuning as "sharper" and the standard tuning as "flatter". So if I want to translate C-tuning stuff (which I'm now more familiar with) to bari names, I have to go sharper: I think "what's the next sharper major key (signature)" or "what's the next position in the fifths sequence" (FCGDAEB, the clockwise pattern for the sequence/wheel of fifths). Whichever instrument I've played (and also from choral singing), I've had key signatures so drilled into me that that's the easier way for me to think. So my (C tuning) D shape plays A on baritone: the next sharper key from D is A. If you're reading lyric sheets for uke but still thinking in guitar shapes, it's the same thing: the chord name you see is flatter than the guitar shape you want.
To find the C-tuning name for a guitar/bari name or shape, I do the opposite: it's flatter, and also the direction chords (particularly dominant 7ths) like to resolve: BEADGCF. This reverse sequence is easier for me to remember because the "hard" part spells BEAD; that just leaves GCF, the three keys with the fewest accidentals. When you have to wrap the sequences around, that's when you make the entire next sequence either sharper (first case) or flatter (second case). The more I do this mental translation between bari and standard, the new names or correspondences stick, linked mentally to the size of the instrument and different hand feel (stretch) of the low-position chords, and now I'm becoming more automatically "binominal".
I've also got a simple orienteering scheme for mapping the fretboard for each size, and when I switch sizes, I may quickly run over the mappings for that size. It helps me switch gears.
Another thing that helps me is that I associate each chord shape I know or learn with which string the root lies on (sometimes the root is doubled and lies on two strings). The root for the D (guitar/bari A) shape lies on S3; F (bari C) lies on S2; G7 (bari D7) lies on S1; the same with G (D) except that it has a secondary root on S3; A (bari E) lies on S1 with a secondary root on S4; and so forth. (Most minor chords vary from the corresponding major shapes by just a fret position on one string, and so have the same root string; similarly with most other closely-related chords, like 7ths, major 7ths, 6ths and sus4s, reducing this extra memorization greatly.) Since I can at least remember the C major scale in first position for each size, the position of the root note tells me which shape to apply. If I'm playing a certain shape, I know its root location on the fretboard, and so can put a name to that note/chord. This is excellent preparation for playing in other positions, where I mostly think in relative root movements or shape progressions anyway: each movable shape relates to a first position one, and thus shares the same shape-to-string root mapping.