including the pith probably won't make the lemonade bitter until it sits for a while. you could get all that lemon oil flavor by zesting the lemon, adding the zest to a simple syrup, and then mixing juice+syrup and water together. That's my usual method for making limeade or lemonade. the syrup dissolves well in water, which helps me avoid adding too much sugar and ending up with sludge at the bottom.
The pith is some of the best nutrition in citrus - loaded with flavones like rutin & quercetin, plus it is mostly pectin which stimulates bile flow, lowers LDL cholesterol, and chelates heavy metals and radioactive cesium from the body.