Lauren, what you are describing is the classic picture of a fast or forceful milk ejection reflex, and Gemma's two adjustments are exactly the ones I would start with. Direct answer first: a baby gaining well, with plenty of output, who fights the breast in the first minute and then gulps and splutters, is usually reacting to flow rather than to a poor latch or to anything wrong with your milk.
The reason positioning helps is mechanical. In an upright cradle hold the milk arrives with gravity behind it. Leaning back so your baby is lying on top of you means she meets the flow going uphill and can control it, and it also lets her come off and breathe without losing the latch entirely. Catching the first strong spray in a cloth before latching takes the peak off. Both are covered in oversupply and fast letdown along with why the green frothy nappies often travel with this and usually settle as feeds calm down.
Two things I would gently push back on. Do not start pumping to "get the extra out", because removing more milk tells your body to make more and you will be doing this in a fortnight at a higher volume. And do not assume the latch is irrelevant either: flow problems and a shallow latch can sit together, and a shallow latch makes coping with a fast flow much harder, so the guide to how to get a good latch is still worth a read.
The professional line, as always: I cannot see your baby. If she starts refusing feeds outright, if the weight gain changes, or if the pain or the dread does not lift, have a midwife or a board-certified lactation consultant watch a full feed in person. Forty minutes of a fight every time is worth someone's eyes.