If you can directly correlate the seal failures to the lack of liquid in the pump then I’d suggest looking into a power monitor. Power monitors are inexpensive and easy to install. I’d still want to learn why the dry run condition occurs in the first place to see if it is preventable.
If you want a mechanical seal solution, your options include a double seal with pressurized flush, a gas seal, or even trying a seal face material combination that allows for periodic dry runs. More information is needed before anything concrete can be suggested.
What is the distance to the nearest obstruction from the face of the stuffing box?
What is the shaft size?
How fast is the shaft turning?
Are there any process limitations? (Material restrictions, intolerance of barrier/buffer fluids or gasses)
What is the suction pressure?
Why is the extra heat from re-circulation an issue?
Do you have a seal drawing or part number to reference?
You are right. Re-circulation wouldn’t help. Without fluid in the pump, no fluid could be re-circulated to the stuffing box to lubricate the seal. (Yeah, I forgot about this and had tubing installed - it doesn’t work) If this wasn’t an issue, you could use a heat exchanger in the re-circulation line to cool the fluid before introducing it into the stuffing box. (API Plan 21)