Selecting needed grid points data

Try this in cell D3 of your format_example sheet:
=OFFSET(INDIRECT("'"&A3&"_"&B3&C3&"'!B2"),MATCH(4.75,'2000_jan1'!$A$3:$A$42,0),MATCH(-7.25,'2000_jan1'!$C$2:$AZ$2,0))
Then drag down.

Note, it threw me off that your 'sample' point was 4.5/9.5 which does not exist on your data sheets, so I changed the values to 4.75 and -7.25. If you need to interpolate for points not on the grid, that adds a new wrinkle...

This also assumes your grids are consistent for each day. If not, you'll have to use the same INDIRECT method for the arrays in the MATCH functions.

Well, I don't really know what is different between your example and the real data. If your sheets are laid out the same and you use the same naming convention, then it should work.
One thing you should try is the 'Evaluate Formula' feature in the Formulas ribbon. Use that on the formula I provided so you can better understand how it works. But, basically, I'm using the OFFSET function. The first input is cell B2 of each sheet. INDIRECT was used to build the sheet name from your dates. Then the next two inputs to OFFSET are the row and column offsets from B2. I used MATCH and the lat/long values in the data sheets to find those offsets. Excel's help can better explain how each function works than I can on this forum, so use that as a reference.
Then use the Evaluate Formula for your modified formula with the big data set. See what it is doing and you may be able to spot the difference.