I understand where you're coming from, but it hasn't been exactly specified what the ailment your character has, though. It could be the Yharnam beast plague or something entirely different. And the guy giving the transfusion says that to solve Yharnam's mysteries, you should have some Yharnam blood of your own.

It could just be that Isofeka was giving the player character a reason to stay in Yharnam. If you arrived at a fabled city to find that it has been overrun by a supernatural plague that transforms its hosts into something beyond what they were originally, why you would stay there? Isofeka obviously has an agenda that he wants to set in motion, and what better way than to force an infection onto someone to make sure they stay in the city. A cure is supposedly out there in the city, and he wants to make sure you stay here to find it and to solve the mystery. If it wasn't for him, the player might as well just turn back and leave the city without giving it a second thought? Why risk life and limb over something you're not involved in. That is, until you're essentially forced to become involved.

Not to mention, the item descriptions for the foreigner gear that you start with basically says that you were wearing this when you arrived at Yharnam, but that you essentially have no memory.

Even then, the player characters account of their life before in the Souls games have never been a point of interest to the main story. Just like in Dark Souls I, we have no idea what they were doing before being placed in the Undead Asylum. The only difference with Bloodborne is that we have has confirmation of why the player character has come to Yharnam since the announcement of the game. Even if the player character has no memory (most likely due to being injected with an altering supernatural plague) from being in a coma for weeks, we still know why they came to the city.