It is [typically] elevated E that causes the liver to produce more SHMG. Your E2 at 32 is not high, not optimal either. SHBG does not drive the HPTA but reacts to it and the effects of aromatase. If one's SHBG is low, then low body fat could be a factor.
Changes in your HPTA and resultant T&E levels will then change the SHBG levels. Consider SHBG levels a result and not a cause.
Lower E will lower SHBG. Lower SHBG will lead to increased FT. An An AI will lower E. A SERM will increase LH and FSH which will increase T if the testes are responsive. However, that also leads to increased E as well and that E will also increase SHBG and offset some of the gains in FT that would otherwise be expected to come with the increase in TT. SERMs can have estrogenic side effects for some as they are estrogen compounds that are selective for some receptors, not for others and can activate some others as well. A few advocate the use of SERMs for some TRT benefits... for those who do not have side effects (only one way to find out). But those approaches cannot take T levels to the high end of normal range that seems to be needed in many cases to achieve results. So as this vicious cycle continue it ends up burning out thyroid, adrenals, pancrease because cortisol and dhea lower shbg, as well does inuslin.
When you lower E2, there is less T converting to E, so more FT+TT left. Less SHBG is created which means that less FT becomes TT. So FT increases. And there is less competition from E at T receptors which can block the action of your FT. If you are on TRT, your T intake levels are fixed. If not on TRT, when you lower E, your HPTA will increase T production.
So with TRT and you lower E, you get less loss of T from T-->E conversion, SHBG goes down, FT goes up and the increased FT:E ratio allows the increased FT to get the job done at the T receptors. That is a lot of positive change factors.
If not on TRT and you lower E, you get the above positive factors and an increase in HL, FST and T.
When you lower E, this can help with fat loss and less fat means less aromatase conversion of T-->E.
See a pattern?