What if you made the tiles a bit bigger so they completely cover the groove, and overlap the edge a bit, so there is no crack to seal?
*nod nod*
I'm an all-around pet person and also happen to have BD (among other treasures). And i can tell you from immense, painful experience that unless that grout/sealer/silicone is super perfect and there are no rips/tears/holes/bubbles, things WILL get under there. and those things WILL mold and rot and generally stink. And when you go to wash them the water gets trapped inside and makes you incredibly frustrated. Or, for example, a cricket will wiggle its way under and lay ever so many eggs that make your entire home a cricket-den for weeks.
... ah, well. you get the idea, i'm sure.

I'd really recommend re-thinking this technique. Its a great starting point, but perhaps cutting the lino so it goes over the lip (i have a super pet also, so i know EXACTLY what you're talking about... huge design flaw, IMO) may help with the sealing process. making some kind of lino 'sleeve' is really the best option. using non-stick lino with iron-on fabric tape for 'joints' would make it super easy to make a really squashed rectangular sleeve to slide on and off the shelf.
i haven't done it m'self, but i know other users have just drilled holes in the 'pee track' that allowed it to drain.
best of luck, let us know how it goes!!