Might be ignorance here on my part about charging the KS10’s with a Kirkland charger but if the charger itself accepts USB input, I would think any Li-ion battery power pack with USB output/input would do. Especially if the Kirkland charger is micro-USB input - that’s not a rapid charger (USB PD) which unless I’m mistaken requires a USB-C port. So when you hook up a battery pack with a 5V output to the Kirkland charger, it should negotiate the correct output (or at least an acceptable output) for the Kirkland device. Even a battery pack with a 5V, 2A output capability should recognize if the Kirkland charger were only 5A and < 1A input and delivery the lower amount of current. And the battery pack and the Kirkland charger can’t arrive at a mutual acceptable “contract,” no power should flow. And you can satisfy yourself as to what’s happening by getting an inexpensive USB digital multimeter on Amazon.
The reason that I mention “any” USB battery pack should do (I would hope) for the Kirkland charger, is that you can buy battery packs on Amazon a LOT cheaper than $120 cited for the Phonak Power Pack (perhaps the same is true for eBay). I bought a 25000 mAh battery pack with a solar panel to boot for $40 on Amazon back in February and it has 3 USB output ports so that one can charge several devices at once from the battery pack. The battery pack claims to be able to charge an iPhone Pro 12 or a Galaxy S9 8 times (probably many more times for rechargeable HA’s). You can get a 2-pack of 10,000 mAh battery packs on Amazon for $20 (a lot less bulky than my 25000 mAh pack).
My ReSound charging case has a 2,200 mAh capacity, said to fully charge both my Quattro’s 3x. So a $10 10,000 mAh battery pack can charge my Quattro’s 10,000/2,200 = 4.55 case charges x 3 HA pair charges/case charge = 13.6 HA pair charges in a $10 battery pack from Amazon.
Just to document the inexpensiveness and the performance of the 10,000 mAh device that I’m referring to on Amazon, I’m providing a link. I haven’t bought it myself. It’s popular and highly rated on Amazon and if it works with your HA charger, you could save yourself a bunch of bucks - that’s my main point. https://www.amazon.com/Miady-10000mAh-Portable-Charger-Charging/dp/B07XFBN7HX/