Having looked around there appears to be some 240v towel rails that don't use too much power.
This one for instance is 'only' 60 watts:
S Style Heated Towel Rail White 60W 550x530mm | Screwfix.com
I reckon a 150w
inverter will cope with that easily.
There are a number of ways to connect it, either way will require a 240v 3-pin plug fitted with a suitable
fuse. With a plug fitted (rather than hard-wiring with a fused outlet) you will be able to use it from your van mains socket while on EHU, or from an
inverter when travelling.
The
inverter can be connected manually with a cigar plug so that you have to remember to plug in/out when using it.
You could connect it through a relay that is energised when the ignition is on (not too clever as the
inverter would be powered up when cranking the engine).
Lastly, and unfortunately the most complicated (and more cost unless you already have split charge) is to connect a relay that is energised when the alternator is charging. A sense wire from your existing split charge system would be fine. If you haven't got a split charge system installed then you will need a Smartcom relay or similar.
Whichever route you choose you would need to consider best paractices for cable size,
fuse protection, cable routing, use of mains electricity in a 'wet room' etc.