This is how we and most of the marquee hire industry lay marquee flooring:
Any marquee flooring should have a non-breathable layer down underneath the main surface. This prevents the grass from giving off water vapour that will come back down as condensation and stops the surface getting muddy. Grass is tough stuff and covering it over for a week or so shouldn’t be a problem.
Use 4″ or 6″ nails to hold down the carpet, this pulls the surface tightly and prevents any trip hazards. On hard surfaces use double sided carpet tape.