I think /u//Speculum has given some good basic info concerning events around Rome in the 350s. But I think this question seems founded on a misconception. Let me explain.

Nicaea (325 AD) is an ecumenical council and decisively deals with Arianism (proper, not counting the thousand and one strands of so-called 'Arianism' that follow. Yes, it's an Eastern council, but it's accepted throughout the West and at this time both East and West think of themselves as one church with no division (but language and cultural issues even now are apparent).

By the 350s most of the East is theologically wading through the issues of post-Nicene theological discourse, and this decade sees the attempts of imperial backed bishops to impose a non-Nicene consensus. Specifically across councils in the West (Arles, Milan, Baterrae) from 353-356, western bishops are being compelled to sign the creed promulgated at the council of Sirmium, which is definitely against the use if 'of one being' language. But the West was prepared to back Nicaea and back Athanasius, up to a point, during this period.

So if you were from Constantinople, and a staunch backer of 'Nicene' theology, you would in 350AD probably find a warmer welcome in Rome than you would at Constantinople.

Will Roman Christians tolerate my form of Christianity?

Obiously we don't have sources giving a full picture. But just eight years before, a synod was held in Sardica under the Roman bishop Julius I, condemning Arianism and its followers and confirming Athanasius, the major trinitarian theologian, as rightful bishop of Alexandria. Athanasius had been banished from his diocese by an Eastern synod held in Antioch a year before.

In 350, the Western bishops still followed Trinitarism, but this changed in 353 when Constantius II became emperor for the whole empire (he had only ruled the eastern parts until then). In a series of synods, the Western bishop were forced to adopt Arianism whereas Pope Liberius was exiled.

In 358, Liberius could return to Rome after condemning Athanasius. He is one of the rare example where the bishop of Rome denounced the Orthodox faith. He is also the first in the line who was not venerated as a saint by the Roman church.

Source: Mostly, Giuseppe Alberigo (ed.): The Oecumenical councils from Nicaea I to Nicaea II (325-787), Brepols 2006. (Can't give page numbers as I'm using the German edition)