It'll work using solfege, but avoid fixed do, or you could use simple numbers, 1-7.
By initially 'fixing' the first note you hear as 'do', or '1', all the other diatonic notes can be practised with regard to where they are in the diatonic scale.
I say avoid fixed do, but using the method above is a sort of offshoot from that - making your own 'fixed do'. Using fixed do works easily in key C, but I feel there's no need in early stages to consider what the note names, or keys, actually are. That will inevitably come later, especially when chromatic notes, thus non-diatonic intervals are encountered.
Once you can recognise intervals using the lower note as 'do', (or 1), then try considering, say mi to ti - which is the same as do to so, but mentally thinking in a different key. Let's face it, if you're trying to establish notes by relative pitch, the lower one won't always be 'do'.
And don't forget, relative pitch needs to be working descending, too. Here, perhaps you may need to swap the notes round, so the first, now, is heard as the lower. Intervals seem easier to identify going up - rather like it's easier to count up, or say the alphabet, days of the week, months, forwards...