Hi Julia, welcome to you and Pepper.

do you have a spreadsheet set up?
We can’t give dosing advice unless we can see the BGs on a spreadsheet. It would be unsafe.
We are very happy to help you but need to see the data.
I will say though that being asleep will make no difference to the BG levels.
An 87 a couple of hours after the dose of Lantus is quite low. Do you know what the preshot number was.?.
I would give him a snack of low carb food is and retest in an hour to see it’s not still dropping.
What dose of Lantus is he on?
I know you are very new and all the is very overwhelming. There are a couple of things you can do that will make it much easier for us to help you. You have come to the best place for Pepper. One is the SS and the other is your signature that will appear at the bottom of all your posts and will tell us about your kitty so we done have to look back through all your posts to see. Here is a link to How You Can Help Us Help You. You will find the links within this link.
http://www.felinediabetes.com/FDMB/threads/new-how-you-can-help-us-help-you.216696/