Something is not quite right.
Lead acid cells have a maximum voltage per cell of around 2.15 Volt. Times 6 gives 12.9 volt as a maximum.
For battery charging, a voltage around 14.4 volt is used.
Presumably the radio is SSB, and if so the current drain is a function of the PEP of the radio. Thus the average current drawn from the supply is quite variable.
You say the unit is used 'mobile' and presumably this means in a motor vehicle. If so, then with the motor running, the vehicle system voltage would be over 14 volt and this would relate reasonably well with the 13.8 volt source. Given the power of the radio and from what you say, it seems that you have a separate battery to power your radio. If so, you must obviously charge it somehow.
One matter I notice is the often overlooked effect of the power lead resistance and the quality of the connections. It is truly amazing to see the conductor gauge of the wire used to supply these car audio amps; you know, the 500 watts units.
Can I suggest you do a detailed examination of your set up with the primary goal of eliminating voltage drop in the supply cables and I would consider to run the radio from the vehicle electrical system.
Hope I'm not trying to teach you to suck eggs, but the approach to use boosters etc seems to me to be the wrong approach because the radio is designed (presumably) to operate off 12 battery systems.
If this was your preferred option(to use a regulator), my preference would be to use 24 volt and chop it down to 13.8. With the boost approach, the peak current in the switching transistor is twice the average DC current and could be up to 50 amp going by your figures. Designing switching regulators at this current level is not easy and requires close attention to heatsink design. Your observation about a proprietary booster unit and its inability to run at 100% of the time is EXACTLY what I'm saying. It aint easy; I've done a switching regulator to run a 12 volt water pump from a 24 volt system in a Toyota bus, and the heatsink and the winding design was critical. For a transformer core, I used a yoke ring from a 110 degree colour TV. The conductor gauge you would need would be around 3 to 4 mm diameter and the overall conversion efficiency is rather low due to the saturation voltage of the switching device.
The cyclons are nice batteries and I have used them in a calibrator, but they are a bit too small for your needs. Seven big ones will cost at least $200 to $250 (probably more). I dont think that is the way to go.
Hope this helps.