The Roman government, despite its oppression, allowed most people considerable freedom, on the single requirement that absolute loyalty be shown to Rome and the Emperor. Since this was well-known, it was often used as a scare tactic by people saying something like, "If we/you do that it will be interpreted by Rome as insurrection".
That is, whether accepting Christ would have raised the ire of Rome or not is a rather moot point but it was used by the Sanhedrin in their internal political struggles. [Note - it is always a sign of weakness that an external threat must be used to gain internal unity.]
The fact that Jesus never encouraged dissent from Rome, but instead encouraged loyalty to ruling powers (Matt 22:21, Mark 12:17, Luke 20:25, see also John 19:11, Rom 13:1-7, Acts 4:27, 28, 1 Tim 2:1, 2, Titus 3:1, 1 Peter 2:13-17, etc) did not prevent Rome being used as a scare-tactic against their enemy, Jesus.
Note the comments of the Pulpit commentary:
Verse 48. - If we let him alone thus, as we have been doing hitherto - if we suffer him to do these things - all men will believe
on him, and the Romans will come and take away from us, i.e. from the
Sanhedrin, from the lawful rulers in all matters affecting religious
order or privilege, our place - the city or temple - and the nation,
which we rule through our subordinates and surrogates, but to
accomplish which we shall prove our incompetence if we cannot keep
down all insubordination and hold perilous enthusiasm in check. ...
Ewald, Godet, Meyer, Watkins, consider τόπον to be the city, the seat
of all the power of the nation, spiritual and civil. The nation was a
province of the Roman empire, but the hierarchy was still invested
with great powers. John 11:48