Okies...firstly it's a red LED, not a laser!
...laser mice are different.
Ni-MH and Ni-CD Rechargeable batteries are rated at 1.2 volts.
Alkaline batteries (non-rechargeable, common standard disposeable batteries) provide 1.5 volts.
It could be that using alkaline batteries has damaged your mouse, but I wouldn't think a 0.3v increase would cause major damage, it usually works fine.
Also, you don't need to look for and use specifically 200mAh batteries for your mouse, just because it came with them. The Milli-ampere hour rating is just units of charge (so how long the rechargeable battery will last after being fully charged). So, using a water tank as an analogy, how much the water tank can hold (the capacity), not how fast the water flows out of the tank (or in other words rate). So, in any device, you can use rechargeable batteries at any rated capacity, and it won't damage the device in any way.
However, the higher the mAh rating, the better, because it means that the device that the device will be powered longer, because the batteries can hold more energy due to greater capacity. 200mAh is very poor, and it means you're going to have to re-charge your wireless mouse more frequently, because it will run out of power quickly. Nowadays, rechargeable AAA batteries go as high as 1000mAh and rechargeable AA batteries exceed 2000mAh.
The only thing you need to be concerned about really is whether the rechargeable batteries are Nickel Metal Hydride (Ni-MH) or Nickel Metal Cadmium (Ni-CD). They are different batteries made of different chemicals. If your mouse originally came with Ni-CD batteries, then the mouse charger will probably only charge Ni-CD batteries safely. In this case, buy the highest capacity (highest mAh rating) Ni-CD batteries you can. Otherwise, buy the highest capacity rated Ni-MH batteries you can.
Put them in your mouse, and charge them up using your mouse charger, then test it. Also, make sure that the optical lens on the underside of the mouse is clean and free of impurities. Use a lightly damped cotton swap or something similar to clean the plastic lens. You might find it easier if you open the mouse to do this.
Also try changing the RF channel if your mouse allows you to.