I can't answer your question directly since I never tried to install games through Lutris directly.
but I do use Lutris (currently v0.5.10.1), with the newest stable Wine from the winehq website, and then I pair it with GloriousEggroll runner for Lutris for the small amount of games I replay (Mafia series and RDR2(with RDR2 I have to set Wine to Windows 10 mode as if you leave Wine in the default of Windows 7 the game won't start up)).
basically I use Lutris with newest Wine (from
https://wiki.winehq.org/Ubuntu ; I use the 'stable' one and after that's done I issue 'sudo apt install wine-desktop-files' which gives you the entries in the Mint menu to configure Wine if needed) paired with the GloriousEggroll custom runner for Lutris (i.e.
https://github.com/GloriousEggroll/wine ... m/releases ; currently you would download "wine-lutris-GE-Proton7-22-x86_64.tar.xz" and extract it to "~/.local/share/lutris/runners/wine/" and then Lutris should automatically use it. if not, you can force it on the games shortcut in Lutris by right clicking it, select Configure > Runner Options and under 'Wine version' you can select the Lutris runner from the drop down menu) and then just make a shortcut in Lutris and guide it to the location of the games main EXE file and then play it.
but off the top of my head if you got a game installer you can probably just install the game like usual through Wine (i.e. double click the installer EXE). once installed, then load up Lutris, make a shortcut and guide it to the location of the games EXE file (which will probably be located in this general location... "~/.wine/drive_c/" ) and play it.
NOTE: to make a shortcut in Lutris it's simple enough... start Lutris, click "+" in the top left corner, then on menu that pops up select 'Add locally installed game'. then a 'Add a new game' window will appear at which point on the 'Runner' text to the right of that says 'Select a runner from the list', click that, change it to 'Wine (Runs Windows games)', then over on the 'Game options' tab under 'Executable' you guide it to the location of the games main EXE file (by clicking the 'browse'). you can tweak more if you want to, but that's the basic idea.
but looking at what you posted... it seems that "lutris-fshack-7.2-x86_64" is not installed by default. but you can install that in particular directly by going to Lutris's 'Preferences > Runners' and then scroll down to the bottom and see where it says 'Wine' and to the right of that is two icons you can click, click the one to the right, which should then open a 'Manage wine versions' screen at which point you can click 'install' to the right of the "lutris-fshack-7.2" text (maybe if you manually install this fshack stuff first and then do your installer, maybe it will work(?)). but honestly, I suspect that GloriousEggroll runner I posted is probably better in general as with that I can run a game or two that normally don't seem to work.
for the record... you can install certain versions of the GloriousEggroll runner from within Lutris directly if you want as they are listed as 'lutris-ge etc' using what I described above in that general 'lutris-fshack-7.2' section.
but looking online a bit for that 'error code 256' your getting one user suggested recently (about 4 days or so ago)... "Top right hamburger menu > preferences > global options > disable lutris runtime (toggle this to off aka no orange)" and a user replied saying it worked for them for whatever this is worth.
