Why can't you do a regular "save as" if you need these large pixel dimensions? The jpeg compression is the same.
Web images were traditionally in the 1000 pixel range, and that's what SFW was designed for. Nobody ever needed a 5000 pixel image on the web.
I assume you know what this aggressive compression does to your file, so I won't bore you with that...
The only reason I prefer the 'Save for Web' feature over 'Save As' is that SFW gives me a real time file size for any given compression. Which comes in handy when I'm trying to eek out as much quality as possible without surpassing my file size limit of 5mb. Whereas with 'Save As' I have to guess which compression to use from the preset levels, then check the file size, and repeat if the compression I chose is too much or too little. Though given the issues with SFW, even that might be quicker/easier, unless there is a third option.
OK, then use Export As. This is SFW's replacement, all new code with no size limit, and file size indicator.
The only options available under Export are 'Paths to Illustrator": which saves the image as an ai file and has no compression options, or "Zoomify" which has pretty much an identical compression feature as 'Save As.' No file size indicator, just the compression slider with preset compression levels.
Maybe the file size indicator is something that was added in later versions?
In Photoshop CS 6 there's not yet "Export as ..." and "Save for Web" is - as D Fosse said - a very old module with limitations.
So I don't see a possibility to resolve this problem except to buy a newer version.
You could try GIMP, it's free - I cannot say for sure if it does what you want.
Why not reduce the image size (pixel dimensions) instead of over compressing the image with jpeg compression.
You can still save as a jpeg, but then you could use save for web without so much compression and get a better quality image.
You can resize the images using Image>Image Size then go to Save for Web.
You could also if you don't want to resize the images, make a new document of the same size, paste the old into the new to get rid of all the metadata and then use File>Save As>Jpeg.
The only reason I prefer the 'Save for Web' feature over 'Save As' is that SFW gives me a real time file size for any given compression.
The regular Jpeg save dialog does give you the file size as you move the slider when Preview is checked, and you see it below the Preview checkbox.
I am using it on a 650 MB open PSD
I stopped using legacy SaveForWeb except for animated GIFs. ExportAs is better.
I also use RIOT - Radical Image Optimization Tool http://luci.criosweb.ro/riot/
Be VERY careful downloading RIOT: the v0.5 installer is known to have malware in it - only the portable version was safe to use.
It seems the installer version of the latest v0.6 beta is no longer infected with malware. But download and use the portable versions only, just in case.
RIOT is pretty good, though: much better than the original old SFW.
Another (paid) alternative is PhotoLine: the web export is very close to the old SFW.
Btw, the reason why SFW in Photoshop bugs out is that it is a 32bit application, I believe.