Upvote:2
If you are travelling on a single ticket, then no, you shouldn't need a transit visa.
While I have doubt about the general situation of a Palestinian with a Lebanese travel document:
there is an exemption that is the same on all sites and applies to you:
If your nationality appears on this list, check whether you have an exemption. An airport transit visa is also not required if you: ... have a visa or residence permit for a country in the Schengen Area, the EU or the EEA
Since Bulgaria is part of the EU, your Bulgarian visa should suffice.