You can actually set up a variational problem in which there is a single continuous displacement field. There is a complete formulation of this type (albeit using a somewhat difficult notation) in Thomas Wick’s paper here (see (15)–(17)):
https://doi.org/10.1016/j.compstruc.2011.02.019
The main “trick” I would anticipate needing to implement this in FEniCS would be assembling the fluid–solid interface term on the appropriate interior facets, since it requires knowing which of the "+" or "-" sides is the fluid subdomain, to use the correct normal vector. This can be done using @MiroK’s answer here:
(I’ve been meaning to sit down and implement this for some time, as an instructive example, but still haven’t gotten around to it.)