You can't just "join" two units, unless they are setup to be connected and sync.
If you want a propane or dual fuel generator, I suggesting buying one that is designed that way.
https://www.championpowerequipment.com/product/100264-3500-watt-dual-fuel-inverter/
This one is also designed to sync its output with another unit to allow 240VAC operation.
Split phase transfer switches do not switch the neutral. The Neutral is grounded at the breaker box or service entry. So it is not going to transmit power beyond the transfer switch.
This is the system bypass I generally use. It is unfused, so you will need to have a subpanel with a breaker. It is a three-way switch, you feed the utility power on one side, the generator feed on the other and the middle is the output to your sub-panel or service.
https://www.homedepot.com/p/GE-100-Amp-240-Volt-Non-Fused-Emergency-Power-Transfer-Switch-TC10323R/100171587
There is a 200A version which should cover whole house for most people.
https://www.homedepot.com/p/GE-200-Amp-240-Volt-Non-Fused-Emergency-Power-Transfer-Switch-TC10324R/100150463
Your idea for #2 is very normal for 120V installations. You feed both side of the 240V panel from the 120V single phase source. If you hook up a 240V appliance, absolutely nothing will work, except if it has some portions that are 120V (i.e. lights). This is quite normal on 120V 30A RVs
BTW, there are 120V 30A single phase AUTOMATIC transfer switches around for RVs.