NOTE : As of today, Oct 14th, 2018, the mod has been updated to implement upgrades. This solution is now obsolete.

Couldn't post it as a comment since it seemed to be too long. I hope it'll be interesting.


Untill the mod is udpated by the author, here's what I found to make them upgradable. First you'll need to install the mod Configure Upgrade slots

https://steamhost.cn/steamcommunity_com/sharedfiles/filedetails/?id=1171964288

by Musashi. Open its XComUpgradeSlots.ini which is located at the adress

...Steam\SteamApps\workshop\content\268500\1171964288\Config\XComUpgradeSlots.ini

here you can add the lines
+SlotConfig = (TemplateName=REF, NumUpgradeSlots=#)

where

1) REF stands for the reference of the TLP weapons, given ealier by Trollsa in the comment section
"Here's the list of all weapons:
TLE_AssaultRifle_CV
TLE_Cannon_CV
TLE_SniperRifle_CV
TLE_Shotgun_CV
TLE_Pistol_CV
TLE_Sword_CV

Changing CV to MG (Magnetic) or BM (Beam) would give the tier 2 and tier 3 variants, which is recommended if you want to add more weapons after having bought the upgrades."

2) # stand for the number of upgrade slots you want : 1,2,3,4...

Add one line for each type of weapons and each tier. Note that, of course, there won't be any cosmetic change, but it seems to work. At least for me it works.

EDIT : What you'll add should look like this

+SlotConfig = (TemplateName=TLE_AssaultRifle_CV, NumUpgradeSlots=1)
+SlotConfig = (TemplateName=TLE_AssaultRifle_MG, NumUpgradeSlots=2)
+SlotConfig = (TemplateName=TLE_AssaultRifle_BM, NumUpgradeSlots=3)

+SlotConfig = (TemplateName=TLE_Cannon_CV, NumUpgradeSlots=1)
+SlotConfig = (TemplateName=TLE_Cannon_MG, NumUpgradeSlots=2)
+SlotConfig = (TemplateName=TLE_Cannon_BM, NumUpgradeSlots=3)

+SlotConfig = (TemplateName=TLE_SniperRifle_CV, NumUpgradeSlots=1)
+SlotConfig = (TemplateName=TLE_SniperRifle_MG, NumUpgradeSlots=2)
+SlotConfig = (TemplateName=TLE_SniperRifle_BM, NumUpgradeSlots=3)

+SlotConfig = (TemplateName=TLE_Shotgun_CV, NumUpgradeSlots=1)
+SlotConfig = (TemplateName=TLE_Shotgun_MG, NumUpgradeSlots=2)
+SlotConfig = (TemplateName=TLE_Shotgun_BM, NumUpgradeSlots=3)

+SlotConfig = (TemplateName=TLE_Pistol_CV, NumUpgradeSlots=1)
+SlotConfig = (TemplateName=TLE_Pistol_MG, NumUpgradeSlots=2)
+SlotConfig = (TemplateName=TLE_Pistol_BM, NumUpgradeSlots=3)

EDIT 2 : Note also that the Configure Upgrade slots mods changes the properties of the chosen weapons, as you can see in the .ini. If you don't want them to be modified, just erase the corresponding lines.