In general I've found that mixing backup types (3rd party and native) is not a particularly good idea. Often OK if the 3rd party tool is just a VDI (taking the native backup stream and doing something, typically compression or encryption, to it), not necessarily so OK if it's doing some snapshot backup form

I've successfully had LiteSpeed full and diff backups and native log backups without issue. I have had problems mixing snapshot-type backups with native log backups.

What is the 3rd party backup tool? Have you checked the SQL error log to ensure that there's no switch to simple recovery happening? Have you checked the msdb backup tables to ensure that the 3rd party tool isn't also taking log backups?