So, to resolve this problem, you will need to use wide heart came on that thick piece of glass. You don't need to use the wide heart came on the entire stained glass window, instead you can use it only around that thick piece of glass. And continue using the regular heart came for the remaining window. When shopping for came, it is possible to find wide heart came in lead, zinc and brass.
If this same situation arises when working with copper foil, the remedy is quite simple as well! You'll need to foil one side of the piece, then flip it over and foil the other side, allowing the foil to overlap in the center area of the edge of the piece of glass. Later, when the piece is soldered in the stained glass window or suncatcher, this piece will be as secure as a regularly foiled piece of glass!