I wouldn't risk directly connecting a 9.6v battery to a radio that expects 5-6v.
The simplest solution would probably be the diodes mentioned in the comment to your question, but as the battery voltage dropped the voltage reaching the radio would drop as well, and as the battery got low the output voltage after the diodes might be too low to power the radio,...
Some of both!
It's normal for a linear power amplifier for HF to be something like 50% - 60% efficient. So to make 100W, the radio probably needs 170 - 200 W.
There are other power draws in the radio, let's say about 20W if it's a fancy modern rig with lots of electronics and a backlit LCD.
Your radio is 13.8V nominal, but it's probably rated for 13.8V +/- ...
The rest (of the current) is why your transmitter requires heat sinks (and/or airflow for thermal dissipation). Linear amplifiers are usually not anywhere close to 100% efficient. A high SWR might reduce the efficiency even further.
If it's a transceiver, then the receive circuitry, front panel, and transmit idle bias also takes some wattage to run.