Offering electrical advice over the Internet always makes me really nervous, but . . . you seem to understand what you are looking and how to use a meter, so here goes.
The fireman's switch circuit you ID'd on the Starite can be connected to any control device or safety, so long as ALL devices are connected in series, not in parallel. In series connections, if any switch fails, the circuit is open (infinite ohms). The circuit is 'connected' or enabled only when ALL switches are closed (> 0.3 ohm)
In millivolt heaters, it is absolutely critical that this circuit must be 'clean' and low resistance, since such heaters operate on tiny voltages and amounts of current. On electronic, transformer driven, heaters such as yours, this is not quite so critical, but still good practice.
It appears that the Jandy has a controlled SP-NO (single pole, normally open) relay in the heater position. This circuit is NOT powered by the Jandy (no voltage), but makes or opens any 24v circuit connected through it. It is NOT to be used with line voltage (>100v) but only with control voltages (<26v).
That said, it appears you can simply unplug the short yellow "fuse" jumper, cut it, and splice it using insulated butt splice connectors.
Do it right -- do not use 'bell wire'. Wire like it was 120v with conduit or 12-2 or 14-2 UF, like this
SouthWire Company 13054221 14/2 WG UF Wire 25-Foot.
You can use a terminal kit like this:
Neiko 175 Pieces Solderless Wire Terminal & Connection with Wire Stripper Crimper Tool
Watch out for the quality of the splices -- unless you go to an electrical wholesaler, it may be hard to find high quality American made connectors. Lowes's and other similar stores sell some awful junk in this category, mixed in with some decent products. The Amazon kit I linked *looked* OK, and is probably worth a try, since you'll spend more than that buying a single box of 3M or other high-grade connections from a wholesaler. Do NOT get any connections from "General Tool" -- I buy stuff from them, but have found their small electrical device quality to be atrocious.
By the way, doing it yourself, or in this manner, may well be a code violation in your area. I have no idea what consequences -- if any -- that may have for you.
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