Hi,
Congrats on your baby! That does sound really messy and difficult for you.
I wonder: what has the LC determined is the problem here? If milk is pouring out of the breast your baby is feeding from then something is definitely not right with the latch (as you've figured out!). Did she check for tongue tie? (this will, at a minimum, involve her putting fingers in baby's mouth to have a feel).
It's possible there's more than one thing going on here. Is baby doing alright otherwise?
If you're in Aus, have you called the ABA helpline for some suggestions? 1800-686-268.
One thing that can be really helpful with latch issues is laid back nursing, also known as biological nursing or baby-led breastfeeding. Maybe this is something the LC has already suggested? If you're unsure on it, you can have a look here. It can be very effective with all sorts of feeding problems.
Good luck