Specifically, would a soul know that the body they're being invited into was created via Reincarnate? After all, for some souls, changing species might be a big deal, possibly of the 'rather stay dead' kind.
The soul knows the body before it chooses to enter it
The reincarnate spell states:
You touch a dead humanoid or a piece of a dead humanoid. Provided that the creature has been dead no longer than 10 days, the spell forms a new adult body for it and then calls the soul to enter that body. If the target's soul isn't free or willing to do so, the spell fails. [...]
The "do so" refers to "entering that body" meaning the body created by the spell. Thus, the soul must be willing to enter the specific body created by the spell; this means the soul can decide that it does not like the body and refuse to enter it. The caster could then try again if they so choose, provided they expend another 1000gp of materials.