I've lost my wallet.dat file and forgotten my password, but I do have my public address.
Your public address is useless. You absolutely require the wallet.dat
and the password (or the private key for the address).
Is there a feasible way for me to recreate a private key that matches my public address?
No, it is not possible to recreate a private key from a public address. If it was possible, bitcoin would be fundamentally broken and worthless. Anyone could steal any coins from any address.
Are there any viable solutions? Thanks!
Depending on
how you "lost" the wallet.dat, it might be possible to recover... was it a harddrive failure or something similar?
In any case, without that wallet.dat file, your chances of recovery are practically zero.