Okay, so

if I'd say "First of January" it would be:

Ichi gatsu tsuitachi (一月一日)


would it sound stupid in your ears if I'd say:

Ichi gatsu Ichi nichi

instead of the first phare?

Its not bad to say it, it means the same thing.

however if u say ichigatsu ichi nichi sounds funny so its better to say ichigatsu tsuitachi

it sounds as strange as January 1th for example. but natives will understand what you truly meant.

^^