You would likely contact print your pinhole shots anyway....
Thanks Keith for the response, I am sure I will start trying out these Fuji (instant) products some time in the near future.
However, I think you underestimate the quality of pinhole camera's.

The Zero Image 4x5 delivers an astonishing image quality.

I have successfully made enlargements up to 12x16 inch, and I think I could go even considerably bigger than that.
The amount of detail captured is truly remarkable, due to a very good pinhole. Of course, there IS a "soft filter" effect, but it is not disturbing, actually, I like it.
I am pretty sure many unexperienced people would NEVER even guess it's pinhole, and assume an ordinary camera.
You know what pushes surprisingly well? Provia 400x. There are some examples of that in smaller format at RFF. I don't know if that will become available in 4x5 though.
Personally based on what I have seen and read up to now, I think "pushing" is ONLY of use if you are still in the NON-reciprocity failure arena (that is, for most films, below 1s measured exposure time.
So yes, you CAN push 400 to 800 ISO and make 1/25th of a second to 1/50th of a second gain so that you are able to shoot out of the hand instead of using a tripod.
However, when we are in the ballpark of reciprocity failure, things may be different. If the exposure is so low that almost nothing is recorded at all (these night time pinholes have completely clear skies in the negs!), than there is also NOTHING to PUSH develop. Underexposure to beat reciprocity failure may just not be the right thing for these pinhole issues. No primary silver particles, nothing to push develop...
So I stay with Eddie's advices and up to now the TMax 400 hasn't disappointed compared to old-style emulsion like HP5. 40 minutes exposure maximum is something I can ride out.
Marco