Personally, I would worry about it getting nasty in there.
Also if there is not much grip in the bottom of the cozy,
this could lead to splayed legs. And on a questions of
humidity; would a cozy be not humide enough or too humide??
I wouldn't know. Maybe it would be okay.
Personally, where these my birds, I would not let them raise
a family in there. I can just imagine how nasty the cozy
would get if the babies hatch. I cleaned my Lovie nest box
every week and it still got pretty bad. I can't imagine how
polar fleece or whatever the cozy is made of would look like.
If I were you, I would personally put the nest in with
nestings material at the bottom and gently place the eggs
in. Place some nesting materials in the cage so the birds
can add stuff.
Yes, there is the chance that she may abandon her eggs, but
I feel that rather then having the potential problems of
them being raised in a cozy (cleanliness and splayed leg)
that it would be better to take the chance of moving them.
Even if you give them nesting materials for the cozy, I am
thinking that some cozies do not have a firm bottom (no
plastic sheet in between the bottom fleece). If this is the
case of yours, I am having a hard time imagining the babies
having a good stable grip. There is also the chance,
depending on how your cosy is made, that the chicks could
fall out.
I have changed nestboxes on my Lovies, they were used to
being a a wooden box, but when cleaning the nesting
materials out I dropped the box and busted it and all I had
was a metal one on hand. I replaced with the metal one and
they go right back to buisiness.
If this is a first clutch there may be a greater chance of
abandonment, also it would depend on your Lovies. I used to
co-parent with mine, they did not mind me playing with the
babies at all or fiddling with the box.
I know, maybe my opinion sounds heartless; but for what I
imagine a cozy is, I don't think it would make the best
nest. I would rather have two potentially fertile eggs
abandoned then risk having splayed legged chicks that may be
disabled for the rest of their lives (I have nothing against
disabled birds, but I think when we have the chance to avoid
this, I'd rather take the chance).
They "can" raise the babies there; I have heard of Lovies
raising babies in hats, on the floor of the cage. A women I
knew of let them raise babies on the floor of the cage on
nothing. They were all splayed legged. I hope they found
good homes.
Good luck in whatever you choose. Others will certainly
give their opinions to aid you in your decision.