Why do these tail-sitters not have bigger propellers in their main engine? My attention was called to tail-sitter aircraft, like these two:

 

Convair XFV Pogo By No information - US Navy, Public Domain, Link



Lockheed XFV 1 Public Domain, Link

These aircraft land on their tails but transition to normal flight during cruise.

But what I want to know is, why are the propellers so small? Disk loading is a big part of efficiency. That's one reason why helicopter blades are so big. It is more efficient to accelerate a large mass of air to a slower velocity than a small mass of air to a faster velocity.

As far as I know, the disk loading factor is valid for propellers in horizontal cruising too.

So why aren't the propellers any bigger? They have plenty of room at the nose. No worries about ground clearance. I would think for a helicopter-like takeoff, this puts a lot more strain on the engine and thus efficiency would be even more important than usual.
You're correct that it is more efficient to accelerate a large mass of air to a slower velocity. But then there are other things to consider here:


The attitude control i.e. transition is based on the downwash generated by the propellers (unlike helicopters). Either you should have high speed downwash or reach higher speeds for the flight controls to become operative. Reaching higher speeds in vertical flight would tax the engines further and the higher speed propellers, however less efficient would have been the better choice.
Huge rotors, like in helicopters would've been efficient, but also bring the related problems of blade flapping, lead and lag and droop, especially leading to problems during transition. There are also issues with tip speed, especially for the high speed interceptor role these aircraft were intended for.
One of the design considerations was shipboard (even submarines were talked about) use, conditions in which higher diameter rotors would've been a headache- you have to find ways to fold them, for one.


In the end, use of smaller rotors would've been influenced by a much more mundane reason- availability of powerful engine. Note that both of these aircraft were powered by variants of the same engine- modified Allison T40 turboprops, the only engine rated for vertical takeoffs and landings available- which consisted of two Allison T38s lashed together to run the same gearbox with contra-rotating propellers.

In fact, the heavier Lockheed XFV never made the transition flight, waiting for a higher powered T-40 and did only horizontal flights with long landing gears (which would've precluded using higher diameter rotors- though I'm not sure if that was a consideration).