If I am understanding your situation correctly, when you were hired you had a certain understanding with your department and university about your duties, which included both teaching and research. Since you don't say otherwise, it sounds like there are "standard" such duties in your department. Also there was a standard contract, and by some strange mistake you were given and signed a different, nonstandard contract. After four years of fulfilling your standard duties, the nonstandard contract has been brought to your attention. Do I have all that right?
Now you are asking whether you should "put up a fight" about the fixing the mistaken contract. If your question is a legal one, of course get a lawyer who is an expert in the specifics of your situation.
But are you really contemplating fighting against what you verbally agreed to when you took the job and the job you have been actually doing for the last four years because a contract you signed says otherwise and the difference may work out in your favor? To me that sounds like horrible behavior. If your bank had noticed after four years that they had mistakenly added a zero to your account balance, would you also fight the correction of the error? Do you not have any ethical qualms about this?
As a justification you say that you wonder that if the situation were essentially reversed, correcting the oversight would be "a real fight with HR". First of all: is this guess grounded in any kind of history with the HR department at your university? Have they in fact been difficult with you in the past? If not it seems like you are simply assuming bad faith. Here's what I think would happen if you discovered your contract was all teaching and no research and tried to change it: there would be plenty of red tape, the entire process would take an order of magnitude longer than you felt that it reasonably should and would occupy too much of your time, but the final outcome would not really be in doubt due to the essential goodwill of the various parties, including people like your department chair/head who would come out on your behalf.
Aside from being ethically highly suspect, a decision to fight the change of your contract is likely to earn you the ill will of HR and other administration in your university and -- probably more importantly -- of the department chair/head and other faculty members. You are contemplating not doing your share of the departmental work because of some technicality that you wonder whether you might be able to get away with. To me that sounds like you are contemplating whether to reserve the right to try to screw over your colleagues at some later time. Unless that is your actual goal, I would steer well clear of this.