Your TT and FT are not extremely low, but definitely low enough to damage quality of life.
With guys on TRT with high-normal TT and FT, E2=22pg/ml seems to be optima for energy, mood, libido, fat loss, fat patterns, initiative and socialization. A very few are wired differently and need more E2. E2=34 will spoil the game for most who have high T levels. In your case, lower T and E2=34 makes you estrogen dominant. Any mood issues?
Lack of body hair suggests low T has been long term. May have lost hair on lower legs, a classis sign and perhaps shiny skin there too.
Your high E2 with low FT suggests liver clearance issues and liver AST/ALT needs to be looked at. We can rule out effects of meds as you do not use any. And we can rule out increased T-->E2 aromatization from abnormal amounts of body fat as you are very lean. So back to liver considerations.
At age 43 there is a great diversity in T levels, so there is no "should ... have".
If you get on TRT and become very sexual, will your wife keep up? This can be a source of tension in a relationship.
I think that you should pursue this issue vigorously. But most doctors are going to reject you. Please read the suggested stickies, this is the best thing that you can do and you my make some connections that we cannot in these posts.
Do get the oral body temps I requested, check your wife's too. If you have not been using iodized salt, there are expected problems. Thyroid hormone fT3 regulates your metabolism by controlling mitochondrial production of ATP, the source of energy for your cells - a strong possible factor in "weakness". Vit-D3 is also very important and if you do not drink milk, get lots of sun exposure or take vitamins, you are deficient. Vit-D3 is converted to Vit-D25, an essential hormone, that oddly is anther steroidal hormone derived from cholesterol.