You might be able to change out your sheeves, 3/16" line is very small. If you go with Spectra it will be strong enough but will be hard to handle. 5/16" is about as small as I would want to use.
If you cannot change out your sheeves you can go with "tapered" halyards. That is one where you have a high tech core like spectra and have a cover on it part way up. This allows a larger line at the deck where you are handling it and the 3/16" core will go though the sheeves.