How much solids should he be eating?

I know there are no rules but...

DS has been eating solids since about 4 months. He is BF and eats three times a day but hasn't progressed beyond a few teaspoons per meal.

He's not interested in the lumpy stuff and his food has to be sweet. I'm hoping to wean him on to formula in a month or so as he is starting to really drain me, day and night!

Should I worry or just wing it and hope his appetite increases? He's definitely not underweight.

Personally I would just wing it. A few teaspoons is still pretty good.
I think before 8ish months they get most of their nutrients from breastmilk (or formula), which is why you are meant to feed the milk first. Eating food is mostly for practice.
If he's gaining weight nicely I don't see any reason for panic