You just need to glue a wood block in there. No big deal. Make a 1 1/2" disc out of 1/2" Baltic Birch plywood. Drill a 1/16" through the center. Drill a matching hole in the butt of the uke. Thread through an old uke string. Thread through disc and tie knot or tie on a bead. Slather on some 5 minute epoxy, and carefully drop it through the sound hole. Pull string tight, pulling glue coated disc into perfect alignment. Hang on for five minutes. Re-drill hole through uke and disc @ 3/32". Install a metal strap button with screw. No problem. Don't try to use a wood or plastic tapered end button...you can't count on the glue without there being a screw in there given that you're dealing with a plastic uke. AllParts has wonderful metal strap buttons. Most people only use a button at the end of the uke and tie to the peghead.
I've just installed a button using a method based on the above by Rick - Rick, thanks for your post, really helpful. I now have a working button and strap on my Dolphin. Whilst not noticeably out the button isn't perfectly perpendicular to the body as, somehow, the holes in the body and in support disc didn't end up in complete alignment. If I did this job again then I'd reduce the disc diameter by a 1/4" to 1 1/4" (to reduce any chance of it fouling on anything) and I'd pull up the disc differently - it should have been fine for me too but for some reason(s), that I can only guess at, it wasn't.
For information I drilled 1/16" holes in the Uke body and the disc and used Araldite Rapid glue. An old C string was used. A slower setting glue could be better as there would then be more time to correct any issues that you might find.
Two alternative pull-ups come to mind:
1) Instead of an old string use a Bicycle spoke instead. Insert the threaded end into the lower bout, push into body, put disc onto spoke, add spoke nipple with plain side to wood, pull wood up to inside of lower bout, suspend from bent end of spoke, check the angle of spoke to body (for Screw hole alignment) and adjust.
After the glue has set drill a clearance hole in the lower bout for the button's screw and enlarge the pilot hole in the wooden disc (if necessary) to suit the screw. I'd use 1/16" pilot holes again, other sizes might work.
2) Alternative - a pal in my Uke club did something similar. Drill the disc to suit the screw and insert to preform form a thread that will be reused later, remove screw. Drill the lower bout with a clearance hole for the screw. Thread some cotton thread through the lower bout and out of the sound hole. Thread the cotton through the disc and secure to a short pencil or dowel behind it. Pull the disc up into the body, align the holes, insert screw and either fully it partially tighten - 'obviously' the cotton is in the way but it (should) gets broken releasing the pencil or dowel. If not fully screwed home then adding a thread and suspending in a way to keep the the disc pulled-up is desirable.
I hope that the above helps someone at some point.