This year I've only had to remove one tick from Barnaby, but in the past it was a daily ritual with Betty.
Very occasionally they don't come out cleanly and on those occasions we've just done the usual of applying disinfectant to the bite.
Of course there is a risk of Lymes disease, but I reckon that if already attached, the disease risk is already there. Unless your attempts at dislodging the tick made it regurgitate while still attached, the risk of disease is not increased.
Just keep an eye on the skin around the bite for a few days. A unusual discoloration/ radial or concentric would be cause for concern and a visit to the vet.
I'm guessing that Eze is blissfully unaware of your worries though:-)