For those running the Galaxy client, here is how to fix LSL7's "999.pal resource is missing."
I'm currently running in Windows 10. So YES YOU CAN.

For the more technically savvy. Need to have a basic understanding of the Windows file system from the command prompt. This is the quickest and most sane way to fix this issue.
Step 1 - Run the installer normally and install the game. (Your most likely already past this point if you were inclined to search for a fix.)

Step 2 -Open up a command prompt. (Windows Key + R and type cmd)
First go to the drive that contains your GOG Games. type "dir /x" and search for your directories 8.3 name. Should be like "GOGGAM~1" Confirm what the directory name is and go into this directory. Do a "dir /x" again. Try to find your LSL7's 8.3 name. If it's the only game installed it will be "LEISUR~1" Do a "dir LEISUR~1" to ensure this is the correct folder. You should see a LSL7.ICO file in the directory if it's the right one. Armed with the short names you can continue to the next step.

Step 3 - Go into the LSL7 folder and edit the "RESOURCE.WIN" file. Quickest means is to run notepad and drag and drop it into Notepad. Change all the paths references to "C:\GOG Games\Leisure Suit Larry 7\" to your short named version which will probably look like "C:\GOGGAM~1\LEISUR~1\"

If all went right... it should be working now. Explanation: A lot of games from the Win95 era were still using the 8.3 standard. Windows 95's long file names were still pretty new when this game came out. They were probably using old file access routines (Win-3.x) which works but has no means of handling long file names. They probably weren't too concerned about this either since the same install also used to have a DOS exe.
If your still having problems it could be from file permissions. Simplest solution is to go to your GOG Games directory and explicitly assign your user account to the security list with full access and tell it to apply to all files and sub-directories.

For the less technical (still need to know basic file manipulation in UI) ... but is more steps and takes longer. Basically we are going to change the environment to match what the game wants.
Step 1 - Go into your Galaxy config (gear in the bottom left corner of the client.) Change the "Install game to:" by clicking the "..." button. Select the drive you want GOG Galaxy to install games into and click "Make New Folder" button. Make a directory "games" Click "OK" button and than exit the Galaxy client.

Step 2 - Assuming you already have LSL7 installed (and probably other games) you will need to go into your old GOG Games directory and move all the files to your new one which is now something like C:\games. Once you have moved all the files you can delete the original "GOG Games" directory. Go into the GOG Games directory and change the "Leisure Suit Larry 7" to "LSL7" Don't close your file explore yet!

Step 3 - Hit key combo "Windows Key + R" and type notepad. With notepad open drag the file "RESOURCE.WIN" to notepad. Change all references to the old path "C:\GOG Games\Leisure Suit Larry 7\" to your new path "C:\GAMES\LSL7\" using the replace option in notepad. Save the file and close

Step 3 - Reload Galaxy client. At this point all your previous installed games that were installed are now broken because they are pointing to the old folder. Select LSL7 in Galaxy client. Click the "MORE" button. In the menu select "Manage..." and in that sub-menu select "Import Folder" Select the new LSL7 folder. And re-import the game into Galaxy client. You will have to re-import all your previously installed games the same way. If this option is not available. Try clicking "MORE" -> "Manage..." -> "Uninstall" and than do the import.

You should now have a working LSL7 install into your Galaxy client.