What have the weight gains and nappy output been like?

Feeding frequently can certainly be normal and there's no reason you can't put him back to the breast again an hour later, rather than the bottle. it's really up to you.
In some cases babies have upset tummies, feed more, then get more of an upset tummy, feed more and end up with too much milk.

Have you just seen the one LC? If she's out of ideas, perhaps you could try another?