How do we know that spacetime itself is expanding rather than some cosmological constant spreading matter?

The Hubble constant predicts that the farther away a galaxy is from your vantage point, the faster it appears to be receding from you. And this in fact is what we see. This would only be possible if space itself were expanding between the galaxies. A force operating within space would not show the same relativistic acceleration over such long distances. Also there is no theory that would account for a force able to expand spacetime in such as way as to mimic the actual expansion of space. What would be the carrier for this force? The two main (and competing) theories that explain forces, Particle physics and String theory (or 'M-theory') do not appear to predict an 'expanditron'. Since Relativity does predict the expansion of Spacetime, and it's been pretty reliable so far, Occam's Razor would probably say the smart money is on spatial expansion.

The alternative to an expanding space is an expansion within a pre-existing space. However it is subject to certain constraints, in particular that no galaxy is 'on the boundary' and that no galaxy is 'at the centre'. This has been done a long time ago and corresponds to the Milne universe. But it does not predict the Cosmic Microwave Background radiation and fails when confronted with modern data., in particular it does not predict the observed relationship between distance and red-shift.. It can be perfected however, but so far with mild success. And there is more: this model was build to deny the role of gravitation in the shaping of the geometry. Galaxies are test bodies without effect on geometry. This is an 'empty' universe from the point of view of a metric theory of gravitation. However general relativity has been tested since in so many ways that it is no longer a tenable viewpoint. Modern attempts at reviving Milne's model incorporate the effect of matter.

The mechanism by which time and space continuously unfolds within the singularity is unknown.

Every point in the universe observes itself to be the oldest and most centralized point in the entire singularity. Even though time and space are continuously unfolding within the singularity, the universe still looks and acts like the singularity of its origin.

Given the data we have collected, where space itself is expanding, your idea is highly unlikely.