I'm Ashkenazi and I usually pray with a Nusach Sfarad or a Rinat Israel siddur, but I have a pocket siddur that I use for travel that is Sefaradi. Can I say birkot hashachar in the Sefaradi version, and then say the rest with an Ashkenazi siddur in the synagogue? What about praying Shacharit with a Ashkenazi siddur and Mincha with a Sefaradi siddur?
In general one should not change once nusach, especially if you follow a strong family tradition (mesorah). Various poskim discuss the possibility to change once (incl. the need for Hatarat nedarim [annulation of vows] in some cases) but nowhere do you find people praying according to different nusachim during the day.
If you pray in a synagogue using a different nusach from yours, you should say private prayers according to your nusach and the public parts (e.g., kedushah) according to the synagogue's nusach).
See various sources and detailed study here according to R Eliezer Melamed in Pninei Halacha, here from him as well in short, here and here.
So it looks like you need a new travel siddur and of course CYLOR for your specific case.