Answer to your questions.
RE: Is there a Biblical injunction against priests marrying?
NO.
is this another man-made rule?
YES.
Plus, it's an option. The Latin Church has opted for celibacy. The 21 Eastern Catholic churches have not. So it looks like there are plenty of choices... more than enough choices, in fact, for married men who want to be priests.
Plus, in the Western church married men can be ordained deacons. So plainly there is nothing about the Sacrament of Holy Orders per se--- which is the Sacrament that both priests and deacons receive --- which excludes married men.
Answer to your questions.
RE: Is there a Biblical injunction against priests marrying?
NO.
is this another man-made rule?
YES.