Wow. Frustrating. Some thoughts.
What about either connection issues with the 2 batteries themselves (like pushing too hard on an internal connector and shorting out the connector), or maybe tripping the protection? Are they 3.0 volt or 3.6/3.7 volt batteries???????? It looks like there are 3.7 volt and 3.0 volt 16340's.
7.2 volts at 4.5 ohms is 11.5 watts and 1.6 amps. A bit hot. Maybe someone with 16340 experience will chime in here. If a 510/510 connector solved the issue for another user... it's either connection problem, or maybe the added resistance of the connector tipped the scale. IDK.
If lower ohm stuff isn't working, but higher ohm stuff is... that's usually protection. What's worse is that freshly charged batteries are 4.2 volts for a while, so that's almost 1.87 amps and 15.68 watts. If it turns out to be a protection issue, high-drain batteries will help.
I do know that some stack 3.0 volt batteries (special charger too) for 6 volt vaping. At 4.5 ohms, that's 8 watts and 1.333 amps.... more in the normal range. This is probably what you are after, but IDK if it is what you have (3.0 v... good batteries).
Like I said... maybe someone that uses unregulated HV stuff will chime in here with experience.
Also, I saw several negative reviews of a set of 16340's on a vendor's site saying they cut out.
Remember this too: Watts = voltage * voltage / resistance.