I used to have a motorhome which had assorted challenges around the shower/toilet room, I think the more recent ones may be better designed and built but in this case I was surprised how short sighted the manufacturers were. I think I understand your description and from it it seems to me that the challenge is that you cannot do any building up around the base of the tap without reducing functionality, and the leak is right there at the base of the tap. It seems reasonable to fear long term damage to the horizontal and vertical panels, especially if they are of plastic coated chip/fibre board. The surface coating or formica type of material will in itself not come to any harm but what is underneath is vulnerable.
Without being there with you it is not easy to advise on what to do or how to do it. However, if you want to have a go at effecting a preventive process then the materials I would think best suited are plastic car body filler, hybrid sealant and Hammerite paint (white ?). The former is a two part compound which sets after a short time so you would need to be pretty clear on what you want to end up with as to a shape (don't mix too much of the stuff at once) - you can build up in stages and it sands pretty easily. Paint the finished shape(s) and use the sealant at the tap. You could try the sealant on its own if you think that might do the trick - mask off to set the limits neatly and have a piece of plastic tool/scraper ready cut to the shape you want the bead to end in (cut from a fairly rigid plastic container).