Static grass doesn't have enough weight to get into the caulk, you'll have to press it down as the caulk is drying.
I've experimented a bit with a couple of options, trying to re-create the mats I used to get from TheTerrainGuy. Whatever you use, cut the mat bigger than you need, pull it taunt and fasten it down. If you don't it'll curl on you. Latex caulk/paint won't take paint and will smell like strong vinegar as it cures. Acrylic caulk/paint is supposedly less durable.
I had a decent success using gesso and Woodland Scenics snow and flock.