Wooden-bodied planes were very commonly oiled (sometimes heavily) with linseed oil and the same could be done with the body of a transitional plane. Many transitionals restored by present-day users are oiled, sometimes on all surfaces.
Apparently transitional planes were sometimes originally sold with a film finish on them instead of oiling1, and despite how unpopular this idea is today with some users a film finish is greatly superior in preventing any tendency towards warping, as well as in protecting the surface from getting grubby with handling and use. Shellac or varnish would have been what was used historically, and either is still a viable choice. Today we have lacquers now too and they'd work fine instead if preferred.
Note that if you use a film finish it should be confined to the upper surfaces, and not used inside the throat or on the sole. Users would then have lubricated the sole as their training and preference dictated2.
Wax seems likely to build up and transfer to the workpiece
No more than when used (as it is very commonly) on the sole of a metal plane.
While it might seem paradoxical, the oil or wax that must inevitably transfer from the soles of planes doesn't seem to affect finishing or glueing. But perhaps we shouldn't be surprised as only a very thin smear of oil is commonly applied, or in the case of wax sometimes just a wiggly scribble down the centre.
but the wooden base probably needs something to prevent it either absorbing moisture or drying out
The reason to do something to the sole is as mentioned to lubricate.
As commonly stored, the sole of a woodie or a transitional would not be subject to taking on or losing much water vapour because it would be resting on a solid surface.
1 Clear evidence for this in surviving planes in very good or untouched condition.
2 Oiled with linseed oil or 'sweet oil' (possibly whale oil), greased with tallow or something similar, or waxed using beeswax.