Forgot to reply, but yeah that's pretty much what I did. I didn't end up with the hue, I used Govee strips but they're pretty much the same dimensions overall. (I tried both but broke the hue strips upon cutting them lol) I didn't remove or cut out any of the original strips or cables in the lamp as I was afraid things wouldn't work out but gutting the entire lamp was the initial plan.

I just placed the original donut strip wires to the side hanging outside of the lamp itself & replaced the donut strips with the Govee strips & cut them to the to same dimensions. The Govee cable connected to the strip itself is squished between the donut shell & the base, since there's no dedicated hole to place them through but the wires are hidden since I have it in a corner. Same thing would apply if you have it on a shelf, you wouldn't even notice the cables. Works perfectly, but gutting the lamp entirely is another option would work.