Are the (presumably identical) BMS units within the "12v" battery packs programmable, or even viewable via cellphone/bluetooth?
The "going high, had to stop it by hand" battery pack BMS could properly be allowing 3.65 max per cell, which could "reach" 14.6V if the cells remained in good balance via the "active balancer". But the "stopping at 13.6V total" pack as an average cell voltage of only 3.4V, and the balancing "start voltage" might be at or above that voltage within the BMS, causing balancing to become active for only a very short time before the charging CMOS is disconnected.
Depending on the kind of BMS it has in it, it most likely contains one "weak cell" going to high voltage (the maximum per-cell voltage of the BMS) before the BMS has had any significant time to perform cell-by-cell balancing. Many BMS units, including nearly all Daly "Smart" BMS units, provide for only passive, low-current balancing between cells. On my two small home-built 12v packs with Daly "Smart BMS", It takes multiple days for the cells to fully balance after a charging cycle, and balancing can only take place at voltages where charging is still allowed.
13.4V total voltage on the "12v"battery pack is about 90% SOC (it/s more than 80%), but I "feel" that storing at that voltage should create no significant issues in total lifespan. Before days of balancing, the individual "weakest cell" is at higher voltage, and it's somewhat in trouble to begin with.
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Are you always going to use the 2 battery packs in Series, as a "24V" battery bank?, In that case, I would remove the BMS from each "12v" pack and re-wire into a single 8-cell pack, electrically speaking. (You can keep 4 cells in each separate case, but instead of being managed by 2 "4S" BMS units it would be managed by one better "8S" unit. The interconnection between the cases would ALWAYS need to remain present, and they would always need to be adjancet to the BMS.)
Upon removal of the current BMS leads, they would be replaced by 9 leads attaching the end terminals and bus bars in sequence (at least 4 being "quite long" to reach into the second pack - presumably bundled within a single pack-to-pack "balancing leads cable"). The inter-pack cable is associated with a balancing lead on the "highest cell" of the first battery. If you want to do this, I strongly recommend this BMS (JK-B2A8S20P, in either the heated or unheated version)
https://www.aliexpress.com/item/3256804034358347.html. (Disclosure: I received the heated version for free, in exchange for comprehensive testing which resulting in the test report
here. I am not associated with the manufacturer or any selling vendor in any other way, and receive no benefit from sales of this unit by anyone.)
That BMS is cellphone/bluetooth viewable and configurable, it also has active balance current of up to 2A continuous. It would solve your issues, and turn you current "balancer" into a spare part.
However, your current single pack refusing to be charged above 13.6 total volts DOES indicate a likely problem with either one "weak" cell, or with the BMS. If the cells are not > 133Ah each, getting only 90% of "full" rated Voltage probably means that your pack can't truly have a working capacity of 120ah.