In my lab we also remove the shell. If possible (e.g. in small organisms) we simply crush it and remove it with forceps. If it is too massive, we put it in a vial with acid.
in my bachelor thesis I weighed the fresh and the dry mussels and burned the organic matter in a muffle furnace and weighed the shells. You can subtracted the shells from the dry mass to get the biomass.
I agree with the responses of colleagues, biomass (dry or wet) is estimated without the weight shell. We do this in our laboratory to estimate the index condition.
Definitely without shells !! Dry and furnace is a good idea. Then you get both the SFDW and AFDW as you burn the "meat" out of the shell. If you are doing a lot of investigations in the area it is worth doing a table and equation of WW/DW/SFDW AFDW for the dominant species of different sizes. General quotas are compilated of e.g Rumohr et al 1987 (pure gold at least for Europe :-). I add a scanned version. Good luck
I agree that biomass is generally defined as an organic weight (fresh, dry, C,...), that is without shell, but it will depend on the scientific question. Shell shape and weight could be interesting if your studies have a link with climate change and ocean acidification
It all depends on the scientific context! If it relates to the organic biomass or in most issues on bioenergetic conversions, ash-free dry weight or biomass as organic carbon would be suitable. However, the build-up of the shell also is linked to metabolic costs. For fresh weight the shell is definitely comparable to for example our bones; it is just an exo-skeleton (compared to our endo-skeleton). Further, the shell of most molluscs also contains organic compounds. So again, it all depends on the scientific question!