We really need to see a schematic, or at least a thorough description of the 7805 regulator area & particularly its decoupling. But you'll be wanting a decent sized electro cap on the input to the 7805, & a pair of ceramics of probably 1.0 to 10uF (check datasheet of specific 7805 in use) on both ~8V input & 5V output, at the very least.
Those batteries might be OK, but I suspect only barely, depending on what kind of load the motors are driving - according to this Alibaba seller's photoed datasheets, they don't even reach their AH capacity at the 20H rate. With no-load=150mA you're at the 10H discahrge rate, where capacity drops significantly, so I'd say there's pretty high internal resistance. Under actual load their terminal voltage might be dropping too low & that's impacting on the 7805 which is in turn putting the MCU out of spec & causing reboots.
Also, drawing power unequally from different cells (tapping off 8V for the control system) is a recipe for premature cell (pack, actually, as Lead-Acid is 2.0V/cell, so theyre's presumably 2 cells in each of those 3 packs) failure if you aren't recharging those 2-cell-packs individually (as distinct from recharing all 3 in series with a '12V charger'), because the 2 cells that are powering the controller will always be at a lower state of charge & get 'flatter' than the other at the 'top' of the series string, and conversely that pack at the top may be being overcharged. Hard to tell without getting hands-on or a lot more info.
Also, there's a discrepancy between that datasheet and your pic - your pic says RB415E (yes? it's a bit blurry) which is 1.0AH, whereas that model# on this datasheet says it's a 1.5AH. Given the shirty specs (not even making claimed AH rating at 20H discharge rate), perhaps they've wisely derated their claimed capacity to 1.0AH & this is just an old datasheet. Who knows. This kind of shuzbut is par for the course on Alibaba stuff :-/