Semi-related follow up: Is it correct to refer to him as Christ in a historical context? I would think historians would prefer 'Jesus of Nazareth', no?

tl;dr: All seems unlikely, but it wasn't unheard of.

Plautus and Plutarch are the two main sources for criminals carrying their crosses. However, two sources does not make a particular strong case for what the norm is. Let's start with what is said by them:

Plautus (3rd-2nd c. BC) is quoted by Nonius (4th-5th c. AD):

Patibulum ferat per urbem, deinde adfigatur cruci. (Carbonaria)

Let him carry the patibulum (the crossbar a convicted criminal carries on his shoulders to become the crossbar of the cross) through the city, then let him be nailed to a crux.

In Miles Gloriosus,

Credo ego istoc exemplo tibi esse eundum actutum extra portam, Dispessis manibus patibulum quom habebis (359)

I think that in that self-same position you will have to die outside the gates, when, with hands outstretched, you will be carrying your patibulum [cross]. (Perseus translation)

Plutarch (1st-2nd c. AD) says in his Moralia

ἕκαστος κακούργων ἐκφέρει τὸν αὑτοῦ σταυρόν

Every wrongdoer who goes to execution carries out his own cross.

So, if the historical Jesus was forced to carry his cross, it wouldn't be unheard of. Whether that was typical, I don't think we have enough information to say. I tend to be more minimalistic in this regard.