You can print a very thin or thick negative.... but it isn't easy to get an appealing result. At least you won't get scanning artifacts! The problem with very thin or very thick negs is that the tones will be clumped and you'll have a hard time separating them. The reason is that those tones are all compressed onto the knee or toe of the tone (exposure vs. density) curve. Nothing's on the linear portion where there is greatest separation.
Note that some negs can be meaningfully intensified. You seldom gain much, but if it's a very important image then it may be worth a try. Nowadays a lot of people don't bother.
To try printing a thin neg I recommend working with multigrade paper and look up the 'split grade' technique. You will need to do something to separate whatever midtones you have, without blowing your highlights or shadows, and that will probably require some careful split tone printing.
Another suggestion is to go ahead and make a dark (overexposed) print and bleach it. You might get some contrast where you want it. There are ways to control that with the bleach formula.
Another thing you might do first is make a duplicate neg with the largest slope of density vs. exposure that you can, i.e., first separate the tones as best you can on film. I suggest this because you can play more tone tricks with film than you can with paper. Some members here can impart lots of information when it comes to manipulating the tone curve of film with this or that developer.
If you decide to go the digital route, my suggestion is to autolevel and print the negative image onto glossy paper, which you can then treat as a paper negative for a traditional print. You can retouch the paper neg here and there on the back with a pencil. Interesting things can result.
P.s. the reason for the scan artifacts is that Dmin is not really zero for your scanner. A lot of people assume it is, but it isn't... especially if you're not using a drum scanner. The only way to fight artifacts is to multiscan like nuts.
