There are a number of things you can do.
First, it's normal to have that gagging sensation on the first flushes, or even after you do a lot of them.
Something simple you can do to overcome this is first chew on a piece of ginger right before drinking the oil.
Another thing you can do is to deeply inhale the smell to acclimate youe body to what's ahead. You might find that you gag on the first smell but not subsequent ones. Or along similar lines you can drink 1/4 teaspoon of the mix first.
Also make sure your digestive tract is completely cleaned out before you attempt to drink the mix.
If you absolutely cannot do any of the above then...
You're onto coffee enemas
as Steve suggested.
are probably second best to flushes, they will stimulate gallbladder contractions but not as strongly as oil.
There are a number of dietary and herbal solutions as well.
The fruitloopy 80/10/10 graham fruitarians do wind up cleaning their livers out with the diet. If you can handle that diet, I think over the long term it will clean your organs out. But only if you can actually stick to it.
Then there is the mini beet protocol, or one-beet-per-day. Beets are a very aggressive liver cleanser.
Herbal bitters can work for some people. If they are strong enough they can be perhaps as strong as coffee enemas. But these things only work in certain cases, and usually if the congestion is not pathological.
Another suggestion would be to supplement taurine over the course of weeks/months. This has been known to work for some people.