Those cells one can buy with built-in protection circuit are usually meant for single-cell devices, such as led lights and e-cigarettes, and are seldom/never used in series. Also, those circuit usually cuts out at 5-7A drain, usually too low for power tools (unless it's in a high voltage packs (48-80 Volt).
Have you measured the Amp usage of those whackers?
Li-cells can be designed for different usage, and those used in power tools are usually high-drain types. I have a Bosch lawn-mower with a 36 Volt, 2.6Ah battery with 20pc. Sony US18650VT cells @ 1,3Ah each. (2P10S) Each cell can handle 10A continuous discharge, and 1.2A charge, so the pack are good for 720 Watt continuous power drain.
But if one tries to use cells from, let's say Sanyo UR18650A (2,25Ah, max. drain 2.15A) from a laptop, on a weed whacker, you'll ruin the cells fast.
But back to your packs;
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You're sure the last cell are electrically isolated by that thermal sensor? Usually, those sensors are only wedged between two cells for thermal contact... I'm guessing that that 'isolation' are the current sense-circuit, and it usually consist of a piece of thick wire on the circuit-board.
I assume your packs are 1P5S - for a weed whacker, i'd really prefer 2P5S, so that the load on each cell would be half.
If you really want to test your cells on the whacker, i'd remove the bms-circuit board entirely; charge each cell properly, and see how it goes on the whacker. But you should really monitor each cell as you test-drain the pack, and stop the drain as soon as one of the cells fall below 3.5V (no load; probably around 2.5V@max load).
As soon as you fint out the max. Amp usage of the whacker, you can start searching for a replacement bms-board. (which probably are the culprit here...)
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