For my horse that I am riding now? No, I would not give this a try. She goes quite well in a conventional snaffle bridle, standard cavason, added some pole padding just because, this bit.
I don’t think securing the bit via a strap around the jaw would be any more comfortable for her - and I question how much tongue relief there would be when contact with the reins is dropped.
I would perhaps consider this for a horse that had bridle / bitting issues - but I have no issues of that manner with my horse (who I started myself, and have always been very careful about her mouth).