this is good adviceI'd use lime mortar as it deals with heat better. For that brickwork I'd use a flush joint hit with a churn brush when it's green hard.
I stripped off a large inglenook fireplace in a listed building last year and used lime mortar.
and I also brushed it when hard but green
I used a tuck pointer, I would do a flush joint flat with it, dont try a bucket handle it wont work on the variable
despite being really rough with loads of big gaps when pointed this way it came up nice
I used a cheap plasterers mini trowel upside down to use as a pointing trowel -works great in small places
for the perps you can just use a bit of mortar in your hand -if you wear gloves it can push in nice
this guy shows how to get the pointing mortar in

