As already pointed out, modern radios have no tolerance for low voltage so this behavior is expected. This, and the real potential of damage to the radio, prompted me to make a fundamental change in how I powered comm gear.
Having been a ham for a few years, I started out with the "wire directly to the battery, POS and NEG with fuses" plan. That plan originated back when radios were more tolerant.
My current method is using a local battery, but with a charging link to the truck system. By local I mean near the radio. In my case (in 3 vehicles) this means a power distribution strip with Anderson PowerPoles, with a 12v 10AH battery on a 6 inch lead (10 ga wire). All the radios plug into the power strip so are effectively isolated from the truck. To keep the 12v "local" battery charged, there is a fused small gauge wire to the truck system (14 ga or so). Shortening the radio leads is a good idea. I removed my heavy radio cables that did go to the truck battery and simply plug the charging wire into the truck's dash power socket (most good for 20 A). That socket needs to be one that powers off when the truck is off so as to prevent feedback into the truck when truck voltage is lower. Optionally, add a diode to prevent feedback, but that will reduce charging voltage by half a volt or so.
Peak charging voltage on the three trucks reaches 14.4 to 14.8 vdc for part of the charge cycle so the "local" battery gets topped up nicely.
Another option is a voltage stabilizing capacitor like sold by MFJ (model 4403) and others. The capacitor would need to be wired near the radio. This is a simple fix for the original problem but does not provide operation with radios disconnected from the truck.