2 Tim 4:16-18 At my first defense no one stood with me, but all forsook me. May it not be charged against them. But the Lord stood with me and strengthened me, so that the message might be preached fully through me, and that all the Gentiles might hear. Also I was delivered out of the mouth of the lion. And the Lord will deliver me from every evil work and preserve me for His heavenly kingdom. To Him be glory forever and ever. Amen!
FIRST, the context is about recent developments in Paul's life - he's just been talking about his most recent trial during which God strengthened him. It is unlikely that Paul suddenly jumps to describing a completely different event without further clarification.
SECOND, the phrase "delivered from the mouth of a lion" also occurs elsewhere:
Ps 22:21 Deliver me from the lion's mouth, and from the horns of the wild oxen.
This makes perfect sense of the use of "lion" to personify troubles or enemies, e.g.
Ezekiel 1:10 There is a conspiracy of her prophets in the midst thereof, like a roaring lion ravening the prey; they have devoured souls; they have taken the treasure and precious things; they have made her many widows in the midst thereof.
THIRD Paul, a roman citizen, could be executed only by beheading, not by being eaten by lions. He would not even have faced such a possibility.
THUS this well-known idiom is used in the context of the proceedings of Paul's (second) trial, and is thus not to be taken literally.
IF you want to argue that Paul faced wild animals in battle, you're better off arguing from 1 Cor 15:32,
What do I gain if, humanly speaking, I fought with beasts at Ephesus?