Hi and welcome aboard.
I think before you ask your GP for T3 you should be asking for him/her to test your ferritin, iron, B12, vitd and folates as if any, or all, of these are low in range they will be having a big impact on how your levo is working, or more to the point, is not working.
Are you keeping your levo, at least, two hours away from any food or drink (except water) and four hours away from any meds or supplements you may be taking.
Have you got any recent blood results you can share with us, if not can you ask the GP's receptionist to print them off and post them on here for advice.
I made the mistake of adding T3 and it caused me heart issues - not because there is anything wrong with T3 but because I didn't actually need it. I, like you, was feeling absolutely awful on levo so talked my GP into trialling me on T3 when all the time it was low Vitd, low iron and very low ferritin that was causing the problems, so I was taking T3 when I didn't actually need it which then lead to heart issues. Without these vital vitamins your body will be unable to convert your levo (T4) into the active T3 it desperately needs so your symptoms will remain.
If, when you have all these tested, they all come back at optimal levels for a thyroid patient (which is a lot different from NHS optimal levels) then T3 could be your answer but T3 also needs all the above vitamins at good levels to work so I would advice that you get your vitamins checked first and then go from there.