It depends on how the dual batteries are wired.
Are they paralleled always? Or are they separated at times?
If separated, is the second battery hot in crank? Or only hot in run?
Is the second battery for running auxiliary loads while the engine is off? (big stereo?)
If so, and if both batteries are hot in crank, then there could be a problem. But that problem would exist anyway if you discharged the second battery (even if an identical battery) all the way down to 11 volts during a tailgate party.
Bottom line: You CAN run two entirely different type of 12 volt batteries if wired correctly.
1. Use an automotive grade, continuous duty, high current rated, diode suppressed solenoid, not those isolators they sell at stereo shops.
2. Wire the solenoid so that it parallels the batteries only in RUN.
3. Do NOT wire the solenoid so that it parallels the batteries in Start.
If the second battery is severely run down, then your main battery won't try to instantaneously equalize it's own charged potential with the uncharged potential of the drained battery (huge current rush) while at that same time trying to start the engine.
Get the engine started first, then while the alternator is spinning, bring the second battery back online to get charged. (This all happens instantaneously).
GM offered dual batteries in some gasoline powered Tahoes... like the PPV police Tahoe. GM used two different sized batteries in that package in 2003-2006 (the only years I know about). GM wired them to avoid problems with using two different sized batteries.
It is not a problem using two different size batteries. OEM's do this all the time. Look at motorhomes with deep cycle "house" battery banks, and then a separate, differently sized engine starting battery. The problem is not having two different sized batteries. The problem is the installation practices in the aftermarket.
The diesel pickups need two batteries to start the engine in cold weather, and those two batteries are paralleled in start, because they are needed at START, and thus are exactly the same size, and should be about the same age, same chemistry, and same ampacity.
People that add second batteries to their gasoline vehicles following the diesel installations as a pattern for the wiring are not doing themselves any favors. If the gas engine doesn't need two batteries to start (it doesn't), then why wire it in such a way that you can't take full advantage of being able to deplete the auxiliary battery without worrying about bringing down the main battery at start?