I breastfed my daughters for two years each, even though I knew I was trans about six years before the first one was born. The only way I could manage that was by disassociating myself from what was happening. I've always wanted my own children and this was the least complicated way to do it with the equipment provided. That's how I stayed sane through six years of pregnancy and breastfeeding. But my dysphoria has become much worse with age and I don't think I'd be able to cope for so long if I were to try it now.
You have to do the best you can and you might need to make some compromises if you're dreading nursing him. Think about this from a different angle: I've seen several scientific studies suggesting that the stress hormone, cortisol, is passed to the baby via breastmilk - and chronic exposure to higher cortisol levels can cause various long-term developmental issues for the child. So if you're getting stressed out every time you need to feed him, he'll be getting stressed too and that might negate the positive effects of breastmilk.
At 14 months I daresay your toddler isn't feeding more than 3 or 4 times a day and that it's more of a top-up than an actual meal as he gets most of his nutrients from solids. You could gradually reduce the number of feeds, one at a time, and see if that helps him wean himself. This is perfectly natural: if you look at how weaning occurs in other mammals, there is always a point where the mother decides 'enough is enough' and she starts gently discourage her young from feeding, usually by simply walking away when they try to latch on. You won't hurt your son by behaving naturally, like any other mammal would.

You could slowly reduce the number of feeds until he just has the bedtime feed to help him get off to sleep, and see how long you're comfortable with that. That's what we did for about the last six months of feeding. So my kids were weaned by reducing down to two feeds a day, then one before bed... and eventually I used distractions to get them off to bed without a feed. We did this slowly (over six months) so as not to cause distress, and it was
fine. You should see how awesome my daughters are today.

My mantra at the time was "she won't be doing this when she's 25". That helped remind me that the situation was temporary and it got me through some difficult times.
As for your career - you're right: it'll be almost impossible to make a career out of it as a man because many women won't be comfortable with a male lactation consultant. Is there some other way you can become involved that won't necessitate front-line support to mothers? Such as policy, admin, service co-ordination, writing, etc...? There is a lot that happens behind-the-scenes and it's every bit as important.
I'm sure you don't need me to point out that binding whilst breastfeeding isn't advisable as you can cause all sorts of problems, but I fully appreciate that you have to do whatever you can to feel as comfortable as possible.
So... well done for managing as well as you have.