Can I sow seeds in well rotted manure?

It will be too strong. Seeds just need a medium to germinate in, seed compost has less nutrients in it compared to other composts.

manure on the surface can be pushed aside for seed sowing, and then moved back when the plants are a fair size.

If the manure has been dug in then the effect is diluted.

I guess it will depend on what you are planning to grow - it might be ok for squash or other heavy feeders. I have at times resorted to half filling larger or deeper pots with my own garden compost and just topping it off with a little multipurpose if I am running short. I have done this for spuds in pots, squash/pumpkins, beans and leeks in deep trays. I would not do it for expensive F1 hybrid seeds.

My understanding is that mushroom compost is alkaline so this might have an impact. As Nobbie suggests the manure may well have had nutrients washed out so might not be too bad. You could sow a few and see how it goes.

I sowed carrot seeds recently in a raised bed which was filled last year with very well rotted manure (had the consistency of top soil) and had courgettes and squash in it. The carrot seeds have germinated with no noticeable difference to those seeds sown direct outdoors.