One solution to perform backups that may take longer than the default sleep time, that does not require preventing your Mac from sleeping, is using Power Nap which allows your backup to be performed during sleep while connected to an AC Power Outlet (assuming you are using Time Machine to backup). Note that backups can only be performed in this method if the Mac is connected to power. However, this solution will work even during the OP's problem of manually forcing their Mac to sleep.

Power nap can be easily enabled by going to Settings > Energy Saver and clicking the check box next to Enable Power Nap.

For more information visit the link above which refers to an Apple support page.

One solution to perform backups that may take longer than the default sleep time, that does not require preventing your Mac from sleeping, is using Power Nap which allows your backup to be performed during sleep while connected to an AC Power Outlet (assuming you are using Time Machine to backup).

Note that backups can only be performed in this method if your Mac is connected to power. However, this solution will work even during the OP's problem of manually forcing their Mac to sleep.

Power Nap can be easily enabled by going to System Preferences → Energy Saver and clicking the check box next to Enable Power Nap.

For more information refer to the Apple Support article:

• How Power Nap works on your Mac