If your RSI has been caused by your work, then to all intents and purposes you are suffering from an “Industrial Prescribed Disease”, either A4 which is for “cramp of the hand or forearm caused by repetitive movements”, or A8, which is “tenosynovitis” etc. If you have one of these “Industrial Prescribed Diseases” (ask for the leaflet from your DSS office) then you can apply for Industrial Injuries Benefit. Your RSI has been caused by your work, so I don’t feel that your employers can slide out of it quite so easily. It may be worth going for Industrial Injuries Disablement benefit; that way, you may be able to convince your employers you are indeed suffering as a result of your work.
Is your Union suggesting you sue your employers? If not, ask them to refer you to your legal department. Your employers caused your injury in the first place, and if you can establish that for instance, your workload was unreasonable, or they did not make reasonable adjustments to your workplace, or they changed your duties and/or workload without consulting with you, then you may have a claim.