I would definitely take out the plastic plant and get him a nice silk plant instead. Try the pantyhose test on the plant that you have right now..which is get a pair of pantyhose, and if the plant snags the pantyhose, then it's more then likely the plant tearing your boys fins up. Betta fins are very delicate.
Also, the size of your tank and your water changes may have alot to do with it as well. It would help knowing the size and water quality.
Here are some tips for treating fin rot. I know it may not be rot, but even if it is torn/frayed fins, I'm sure this will help him in the same way.
Tail rot or fin rot
•Symptoms: Betta’s fins and/or tail seem to be getting shorter and shorter or they seem to be falling apart and dissolving, Black or red along the edge of the betta’s fins/tail, Bloody tips, Behavior may not change
Treatment: Conservative: Treat with Aq.Salt at 1 tsp/gal. Increase water changes to 100% daily. Replace accurate amount of salt following water changes. Add Stress Coat to help repair tissue. If there is little to no improvement within the first 5 days, you can increase the salt dosage gradually to 2tsp/gal but do not continue any salt treatments past 10 days. Medication: If Conservative treatment is ineffective use API Tetracycline, API Fungus Cure, API Triple Sulfa, OR API Erythromycin. Also add Stress Coat to help regrowth. Continue until fins/tail stop receding and start showing some new growth.
Hope this helps!