When we speak of secondary hypogonadism we only mean the problem isn't testicular in nature; the issues lie somewhere else. The pituitary, some other axis problem, possibly rooted in a thyroid irregularity, or cortisol. That said, your labs suggest that it is secondary hypogonadism. Your LH being low and your FSH being below the midrange means your body is not signaling for those hormones in excessive amounts. Your testicles are fine.
You can't really raise or lower SHBG, you play the hand you're dealt. Injections of androgen typically lower it, and if you were to initiate TRT larger, less frequent doses of testosterone would probably be prescribed.
There are more tests needed before any therapy should start. Have you ever had a full thyroid workup? Not just TSH, but ft3, ft4, rt3, and both antibody panels? Your history suggests issues long in the making; I would certainly zero in on thyroid. Many men complain of TRT failure but never have their thyroid evaluated. I'd also look at prolactin, at a minimum.
