I grill fish outside to avoid the smell in the house. Because fish is so healthy, we do this a lot. I`m beginning to get worried that the grate is not clean enough. My question is: Can I use an "indoor" oven broiler grate, outside, on top of the grill grate, so I can physically bring it in after each use and completely clean it in the sink? That way the food never touches the grill grate.(We have tried several wire "fish baskets" and they are a royal PAIN to clean. Crumbs get stuck between the wires.)
I have a few issues with your goals and current expectations.
Keeping the grill grate clean
Is this really desirable? In my experience, you get a lot of flavour from the burning off of the last meal. It's a delicious ancestor effect. Maybe that doesn't bother you.
But washing a cast-iron grill (my assumption, might not be) is also a bit hard on the material. Cast iron is happiest when it's covered in grease. Deep cleaning it between every use might (citation needed) shorten its lifespan.
Keeping the grill grate sanitary
If you're using it like a grill, it's getting really hot. You've nothing to worry about unless you have commercial food safety standards to uphold. Even those don't require a daily deep clean (I'm sure that varies around the world).
You could follow some simple "best" practice when grilling food.
(disclaimer, I use a Weber Q grill with cast-iron grill)
You want to have your food touch the grill, that is how you get nice grill marks; using a regular oven grill will result in the same issues you have; the only situation I would use a different grill is if wanting to grill smaller ingredients that would drop between the grill.
Pre-heat your grill. When the heat is nicely hot, you a sturdy metal brush to clean the grill.
Use a paper towel dipped in some cooking oil to clean the grill (use a pair of tong)
(re)Heat the grill to proper temperature.
Put fish (or any other food on the grill) and cook.
When the food is ready and served, use the metal brush to clean the grill and let the heat cook down any residue for a few minutes more.
If you want to clean the grill with water and soap, use regular dish-soap and use a plastic brush or scrunge sponge and clean it up and make certain you dry it completely to prevent rust (if using cast iron, you can rub a little bit of oil).
Yes, you can bring an indoor oven broiler and put it on top of the outside grill grate, as the metal of the indoor one will absorb the heat of the outside one...
However, it will take a little bit more time for the indoor grill to have the same temperature as the outside one... ¯\_(ツ)_/¯
Alternatively, use a copper wire brush to clean the grill after you've used it as in this answer here before and after each use while the grill is still hot.