How do I go about sealing tiles painted with acrylic ink?
Both methods should work; polyurethane varnish seems to be a fairly standard way of sealing acrylic paint on ceramic (see this thread on potters.org for example) and Art Resin* claims that 'the heat generated from a hot mug will not damage the resin surface' and that it is suitable for use on ceramics (ArtResin FAQ, points 24 and 25).

I also stumbled across some people saying that you can bake acrylics in the same way that you can enamels - no guarantees, but you might find that this works for you.

However, given that you've already had some issues with the ink dissolving in the varnish(possibly caused by attempting to varnish before the ink was fully dry) I suggest that you test your chosen method on something disposable before you seal your coasters.

*a brand, but the same points should apply to most clear two part epoxy resins