Is there any way to make our own Invite Friend without having default SocialService Invite Friend GUI?
No, you can only use the default GUI.
If Roblox allowed you to invite friends manually, developers could easily exploit that API and spam your player’s friends.
The closest thing I found was to use
PromptGameInvite, but that makes the game invite screen pop up.
There is no real way due to what @Dawgcool13 stated that it could be used to mass spam players to join a game and no way to stop this.
The only real way is to use the SocialService:PromptGameInvite as what @Diamond_Boiiiiiiii stated but this does not allow you to have it custom only the default Roblox game invite UI to come up.