There are a couple of possibilities for why your tap valve washer isn’t coming out easily. The first is that it’s just stuck. Sometimes, washers can grip on the edges of the valve body, and using a pair of needle-nose pliers to gently wiggle it while pulling straight out can help free it.
However, the more likely situation is what you suspected: you have a spindle extender installed. This extender effectively lengthens the valve body inside the wall, and its inside diameter can be too narrow for the washer to pass through. In that case, you are correct. You’ll likely need to carefully cut away the silicone around the valve to access the extender. Make sure to wear cut-resistant gloves and go slowly to avoid damaging anything.
Once the silicone is removed, you should be able to unscrew the extender (usually around 25 mm long). With the extender out, you can remove the washer and replace it. After that, you can reinstall the extender, followed by your spindle and its own little extender. It can be a bit tricky to line everything up, especially inserting the spindle with its small extender into the valve inside the wall, but taking it step by step should work.
Please let me know if you have any questions.
Mitchell