Struggling Building a Garage Maximum 2.5m High

The brickie doesn't like the idea of having a fall visible from the inside of the garage. He prefers a firring above the joists. I was being over cautious with the 1-40 fall as that allows for deflection and errors. To have a 1-80 fall as is the minimum, it's circa 69mm plus 18mm ply so circa 90mm above the joists.

If we ran a universal beam (RSJ) across the middle of the span, we could hang smaller joists. I need to work out what rsj is suitable and I don't know how to calculate this. On the basis building control aren't interested as it falls outside of their remit, we're guessing a bit and reckon a 203 x 133 x 25 should do it with 145x47 joists.

So 90mm down, 145mm joists = 235mm which leaves 2265mm. If I used a 152mm rsj as a garage door lintel (2800mm spanning 2500mm), then that leaves me 2113mm for door height which is what I need there and abouts.

Any opinions on this? I'm trying to not go to our structural engineer as we've had some questionable 'opinions' from him so far.
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my thought
remember its above the natural ground level the structure covers not the internal floor level
i would think off sourcing the door first as the frame may be a surprize to the equation
yes the bricky may prefer level and firings but the 50-70mm at the front above the door may be the difference between 2550 fail and 2490pass
The one I mentioned needs less than 100mm, you can see how high up the lintel the frame goes:
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