Definitely do the dog x-pens – those are the best in my opinion.

Wow, sounds like a tiny bunny!  I would also recommend an ex-pen though it sounds like you will have to reinforce the spacing until your bunny is bigger.  I would check out wire fencing at the local home improvement store- and zip tie (cable tie) that to the outside of the ex-pen to prevent any wiggle thrus.  Sometimes the edges of wire can be sharp so just make sure that there is nothing that Godzilla can scratch himself on- and nothing that you will scratch yourself on when you are walking by.  My first thought was to recommend a plastic fencing- but I really think something the bunny can NOT chew thru would probably be a better bet- that way you don’t have to worry about him escaping and getting into trouble when you are not watching.

For underneath the cage you could use a piece of linoleum or something similiar.  I use vinyl from Joann Fabrics- it is sold on a large roll and I think other people use it to cover picnic tables, etc.  It is clear (or tinted) and there are different thicknesses- but it is pretty cheap.  I learned with my first bunny that anything you put on the floor to protect the carpet should lay flat- otherwise they will nibble the wrinkes.  So a tarp if it flat, linoleum, vinyl or tiles is fine to protect the floor.

I cover the vinyl with fleece for my bunnies- so the carpet is protected by the vinyl and the bunnies are pampered by the fleece.

I third the xpens. They are the best in my opinion. Can you pick up some NIC grids and ziptie those around the perimeter? When you no longer needed them for the xpen, they could be used for bunny-proofing somewhere else in the house.