Of course it would be best not to inaugurate your possible future career with this employer by infecting everybody on first meeting. But I can see why you wouldn’t want to cancel.
In your place I would definitely dose myself up with whatever my favorite remedy is for suppressing symptoms without causing drowsiness. Try to take the meds at the optimal length of time before the interview (as guided by experience; I would say maybe an hour) to sustain the strongest effect while you’re actually on the premises.
I’d also do a really good head-clearing in the restroom just beforehand.
Try to seat yourself at as great a distance as you reasonably can, and preferably at an angle, not facing straight across. You can say “I seem to be coming down with a bit of a cold, so I’ll keep my distance.” Definitely no hand-shaking.
And yes, I’d come supplied with tissues and also have a plastic bag or equivalent in an ample purse so I could discreetly stow the used ones and not end up holding them, piling them, or dropping them.
Meanwhile, today I would be doing the whole rest-and-fluids bit, plus vitamin C (I don’t care who says it’s not effective), to try to fend it off as much as possible.
I have found that with concerted effort I can brazen out about three out of four colds that head my way. But it does take jumping on the symptoms right away and treating them with all diligence.
The fourth is usually a doozy.
P.S. If your nose is red, do not wear red or pink. That will really bring out the color of your nose. If you have a nice outfit in please-take-care-of-me blue, wear that.