Looking at the second set of results, your Free T3 is 30% of the way through the reference range and your Free T4 is 19% of the way through the reference range, and neither of these are good results. On this forum, for people on levo, it is suggested that Free T4 should be in the top quarter of the reference range, and Free T3 should be in the top third.
You are under-medicated in my opinion. Neither your Free T4 or your Free T3 are high enough. The problem is that your GP is clearly going to stick to the guidelines no matter how you feel. (You aren't the first patient to have private blood test results thrown back at them.) And with all three numbers being in range you are unlikely to get a raise in your levo.
You have a few choices :
1) Get your ferritin, folate, vitamin B12 and vitamin D tested. If your doctor won't do it then you may have to pay.
Finger prick testing from Blue Horizon is good value for money. See the Thyroid Check Plus Ten. There are discounts available from Thyroid UK on testing costs. See the link I've given and follow the links within it.
Vitamin D testing can be done with City Assays or Betteryou, also with finger-prick testing.
If you have any deficiencies then repairing those may improve your thyroid function test results, and will make you feel better anyway.
Have a look at medichecks as well. No discounts available, but is worth investigating.
I've never used them but some people on the forum have.
2) See a different GP at your surgery and ask for a raise in your levo. I don't know how successful you'll be. Getting prescribed more levo when all results are in range is difficult and probably quite rare. If you still don't get anywhere then you could try changing your GP surgery, but it is a leap in the dark with no guarantees of getting a different result.
3) You could start self-medicating. If you do that then you could just buy some more levo and top up your own dose. Or there are other possibilities - you could add T3 to your levo, or you could try NDT, or you could try T3 alone. This isn't necessarily an easy option and you would need to understand how this all works before you did anything like this.
Edit : I'm not medically trained, so following my suggestions is something you do at your own risk.