If Jesus had claimed to be a god, I doubt it would have gotten him into much trouble, unless he claimed to be the only true god and every other god was a false god. The Romans were pretty tolerant of differing religions. They only insisted they receive the same toleration back. The Christians and Jews were constantly getting into trouble not for having their own religion but for refusing to respect other religions.
Refusing to worship the emperor was later seen as rude and disrespectful. In wartime it could be seen as unpatriotic. That sometimes got Christians and Jews into trouble. It's not much of a stretch to turn unpatriotic into treasonous. There were several persecutions, but I'm unaware of any Christians being 'thrown to the lions' so to speak for a specific charge of treason. During the various persecutions, being a Christian was usually grounds enough to be 'thrown to the lions.' During the last persecution, in the late third century, just before Constantine, Christians were ordered to cease practicing their religion and to turn over their sacred texts for destruction. Christians were sometimes executed for non-compliance with these laws. In ancient times most crimes were punished by death, not just treason.