Thanks to some help from Goob, I've gotten a bit further. I've added code.lib to the libraries for FRED and FreeSpace2, and taken out linci.lib. I now get this error on all three projects:
BSCMAKE: error BK1513 : nonincremental update requires all .SBR files
Also, both the FreeSpace2 and FRED projects gives me this:
code.lib(collideshipship.obj) : error LNK2001: unresolved external symbol "class ade_obj<struct object_h> l_Ship" (?l_Ship@@3V?$ade_obj@Uobject_h@@@@A)
code.lib(collidedebrisship.obj) : error LNK2001: unresolved external symbol "class ade_obj<struct object_h> l_Ship" (?l_Ship@@3V?$ade_obj@Uobject_h@@@@A)
code.lib(collideshipweapon.obj) : error LNK2001: unresolved external symbol "class ade_obj<struct object_h> l_Ship" (?l_Ship@@3V?$ade_obj@Uobject_h@@@@A)
code.lib(windebug.obj) : error LNK2019: unresolved external symbol "void __cdecl ade_stackdump(struct lua_State *,char *)" (?ade_stackdump@@YAXPAUlua_State@@PAD@Z) referenced in function "void __cdecl LuaError(struct lua_State *,char *,...)" (?LuaError@@YAXPAUlua_State@@PADZZ)
code.lib(windebug.obj) : error LNK2001: unresolved external symbol "struct lua_Debug Ade_debug_info" (?Ade_debug_info@@3Ulua_Debug@@A)
code.lib(scripting.obj) : error LNK2019: unresolved external symbol "public: int __thiscall script_state::CreateLuaState(void)" (?CreateLuaState@script_state@@QAEHXZ) referenced in function "void __cdecl script_init(void)" (?script_init@@YAXXZ)
code.lib(scripting.obj) : error LNK2019: unresolved external symbol "int __cdecl ade_set_object_with_breed(struct lua_State *,int)" (?ade_set_object_with_breed@@YAHPAUlua_State@@H@Z) referenced in function "public: void __thiscall script_state::SetHookObject(char *,int)" (?SetHookObject@script_state@@QAEXPADH@Z)
code.lib(scripting.obj) : error LNK2019: unresolved external symbol "int __cdecl ade_set_args(struct lua_State *,char *,...)" (?ade_set_args@@YAHPAUlua_State@@PADZZ) referenced in function "public: void __thiscall script_state::SetHookVar(char *,char,void *)" (?SetHookVar@script_state@@QAEXPADDPAX@Z)
code.lib(scripting.obj) : error LNK2019: unresolved external symbol "int __cdecl ade_get_args(struct lua_State *,char *,...)" (?ade_get_args@@YAHPAUlua_State@@PADZZ) referenced in function "public: bool __thiscall script_state::GetHookVar(char *,char,void *)" (?GetHookVar@script_state@@QAE_NPADDPAX@Z)
code.lib(scripting.obj) : error LNK2001: unresolved external symbol "bool Ade_get_args_lfunction" (?Ade_get_args_lfunction@@3_NA)
code.lib(scripting.obj) : error LNK2001: unresolved external symbol "int Ade_get_args_skip" (?Ade_get_args_skip@@3HA)
code.lib(scripting.obj) : error LNK2019: unresolved external symbol "int __cdecl ade_friendly_error(struct lua_State *)" (?ade_friendly_error@@YAHPAUlua_State@@@Z) referenced in function "private: int __thiscall script_state::RunBytecodeSub(int,int,char,void *)" (?RunBytecodeSub@script_state@@AAEHHHDPAX@Z)
code.lib(scripting.obj) : error LNK2019: unresolved external symbol "private: void __thiscall script_state::OutputLuaMeta(struct _iobuf *)" (?OutputLuaMeta@script_state@@AAEXPAU_iobuf@@@Z) referenced in function "public: int __thiscall script_state::OutputMeta(char *)" (?OutputMeta@script_state@@QAEHPAD@Z)
code.lib(collideweaponweapon.obj) : error LNK2001: unresolved external symbol "class ade_obj<struct object_h> l_Weapon" (?l_Weapon@@3V?$ade_obj@Uobject_h@@@@A)
code.lib(collidedebrisweapon.obj) : error LNK2001: unresolved external symbol "class ade_obj<struct object_h> l_Weapon" (?l_Weapon@@3V?$ade_obj@Uobject_h@@@@A)
code.lib(collideshipweapon.obj) : error LNK2019: unresolved external symbol "class ade_obj<struct object_h> l_Weapon" (?l_Weapon@@3V?$ade_obj@Uobject_h@@@@A) referenced in function "void __cdecl ship_weapon_do_hit_stuff(struct object *,struct object *,struct vec3d *,struct vec3d *,int,int,struct vec3d)" (?ship_weapon_do_hit_stuff@@YAXPAUobject@@0PAUvec3d@@1HHU2@@Z)
code.lib(collidedebrisship.obj) : error LNK2001: unresolved external symbol "class ade_obj<struct object_h> l_Debris" (?l_Debris@@3V?$ade_obj@Uobject_h@@@@A)
code.lib(collidedebrisweapon.obj) : error LNK2001: unresolved external symbol "class ade_obj<struct object_h> l_Debris" (?l_Debris@@3V?$ade_obj@Uobject_h@@@@A)
code.lib(collidedebrisship.obj) : error LNK2001: unresolved external symbol "class ade_obj<struct object_h> l_Asteroid" (?l_Asteroid@@3V?$ade_obj@Uobject_h@@@@A)
code.lib(collidedebrisweapon.obj) : error LNK2001: unresolved external symbol "class ade_obj<struct object_h> l_Asteroid" (?l_Asteroid@@3V?$ade_obj@Uobject_h@@@@A)
Debug\fs2_open_d.exe : fatal error LNK1120: 14 unresolved externals