As a therapist specialising in hand and upperlimb injuries (I treat a huge amount of people with tennis elbow (lateral epicondylitis), if your doctor thinks it's work related - it's well worth making it a workers comp claim. Only your doctor can make the call on whether it's work related.
The treatments can vary, but CSI (cortisone injections) can sometimes alleviate symptoms shot term, but do not always change the long term outcome.
If you lodge a workers comp claim more investigation is usually undergone (ultrasound or MRI) to work out the extent of the injury. Treatment with an Occupational Therapist or a Physiotherapist may include splinting to decrease inflammation prior to an exercise program or an exercise program.
I won't go into what I particularly do here (it is a public forum), but I'm happy for you to pm me for more details.
You can still get all treatment required if you do it privately, however, if you go under worker's comp, your DH is more likely to have someone coordinate treatment and investigations which is more likely to get an outcome quicker. (This can be very dependent upon the insurer claims officer's experience, or the GP's experience, or the physio/OT's exerperience, whether they employ a vocational rehab counsellor, organise a return to work plan, understand which investigations will give more information, perform a worksite assessment etc.)
As I said - happy for you to pm me - but a lot of this stuff is individual to the case and a better answer can occur with more information