Throwing in a diagnostic summary:

1. Lower right side back pain only occurs while riding indoor training.
2. Outdoor, long rides do not cause pain.
3. Bike fit is up to date.
   
4. Bike is level.
5. Saddle is level.
   
6. Bike is the same as used for outdoor riding.
7. Core & leg strength routines do not help.

My thoughts are the anchored bike position (no sway and no fore-and aft movement). Or it is fatigue from staying in the very same position on the trainer, while you might move around a lot more in outdoor rides.

First I would try making a conscious effort to simply move around more on the trainer rides. Stand some, change hand positions on the bars, shift occasionally on the saddle, etc.

Building on this, do you develop back pain riding rollers indoors or just the trainer. For me the order of operations is trainer is most locked down between upper body static hips locked into saddle and feet locked into pedals. On the rollers there is more decoupling between upper body anchor points and hips so psoas and hip flexors more around more relative to shoulders, in other words the entire spine has some wiggle room. Outdoor is better than rollers just because you change position the most. Indoors, the entire spine from cervical to lumbar is relatively locked in.

I make sure I ride "courses indoors" where I sit up a lot and ride out of the saddle. Also just try to ride 1-1:20 rides on trainer that are higher intensity so that there is less weight on saddle just from the intensity. Longer rides on trainer, too much weight on sit bones for too long while upper body is locked into an endlessly static position.

It was fine when I was young and had a bomb proof spine....not so good after 45 years of riding and my FTP is 25 percent lower, meaning there is just more weight on the saddle all the time (when I was riding all the time at all intensities harder, you're more floating off the saddle due to force from legs into pedals).

It would be really helpful to see a pic of you on your bike, on the trainer.

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Tech writer/support on this here site. FIST school instructor and certified bike fitter. Formerly at Diamondback Bikes, LeMond Fitness, FSA, TiCycles, etc.
Coaching and bike fit - http://source-e.net/ Cyclocross blog - https://crosssports.net/ BJJ instruction - https://ballardbjj.com/

I spent a lot of time on the trainer last year and developed a number of issues. I have a Kickr Bike which is even more rigid. If you don't want to spend money on the movement tech, I'd take Dev's advice and stick to shorter rides. You could even break them up doing two threshold sessions a day like the Norwegians.

I've since sold the Kickr bike and just ride outside whatever the weather.