Ok, here are my thoughts on it
I think it's too much of a rush to build a bin cage. I think the large PAH wire cage is a good option for about 3 weeks. It is slightly taller than the medium plastic home and has slightly more floor area due to the shape. It's also a barred cage so it will be less of an unfamiliar style of cage when moving into the Barney.
It's 30cm tall. The medium plastic is only 26cm tall. You'd need to take out the 6" (or smaller) wheel that comes with either the large wire or medium plastic - because it's too small for a Syrian. You'd have a problem fitting an 8" wheel in the medium plastic, but it would fit in the large wire.
The Barney comes with an 8" wheel which would fit in the large wire PAH cage temporarily until you move your hammy to the Barney with a larger wheel, so that would save buying a wheel. As the Barney will be with you in a few days time, the hammy would manage without a wheel for a few days (until you can take the wheel out of the Barney to use) if you can rig up a playpen so she can have a bit of a run around out of the cage. The bathtub would do (dry bathtub with plug in). You could also use a hamster ball (outside the cage) for a while to help with exercise until she has moved, but I wouldn't use one long term.
I think either the medium plastic or the large wire PAH cages would be fine as a starter cage for 3 weeks then upgrade to the Barney. Personally I think the wire one would be better if she is being upgraded to a wire cage, plus a wheel will fit in better. It would also be useful to keep as holiday cage or for a play cage when you clean the Barney.
An 8" wodent wheel (as an example) might just fit inside the medium plastic home. It is 23cm tall and one of the lower overall height 8" wheels. The 26cm height of the medium plastic home is maximum external height and not at the curved corners. It might only be 25cm or less internally, depending on how the top lid fits. So personally I think it would be much easier to fit any brand of 8" wheel in the large wire pets at home cage, plus the Barney one would fit to the bars.
You wouldn't fit a flying saucer in either of them. A Syrian needs a 12" flying saucer to be able to run properly. The smaller ones they can't run on well and it's not nice for them, even temporarily.
So the main issue I can see with the smaller temporary cage is fitting a wheel in. They do need to use a wheel daily, especially in a smaller cage. don't think Pets at home sell any 8" wheels.
The most important thing is getting her settled in right from the start, and then upgrading later. So either of the pets at home ones would allow her to settle in without needing to upheave anything for the first couple of weeks. Whereas trying to rush a bin cage could mean needing to disturb her during her settling in while you need to finish bits or something. On the other hand you may well have made one by now! I think they can be a bit fiddly to make with cutting and meshing taking time.
Adding the wheel from the Barney a few days later would be a positive change. The main thing is, leave her to settle in to whichever cage for 2 or 3 days undisturbed, to scent mark and build a nest and just settle. Then don't clean anything for two weeks. If it gets a bit whiffy you can spot clean after 5 to 7 days - ie just take out a handful of pee'd on substrate and add a new handful and mix it in. Put lots of substrate in then it stays clean longer and they can enjoy digging and rearranging it.
With the wire cage you could also do with a house adding - a tissue box with a side cut out would do fine
The little plastic house is a bit small and also it's best if a house is open underneath sat on the substrate so she can burrow down and bury hoards and feel protected.