Informative list of all saving approaches and their memory-limitations in cordova can be found here.
But when you really want to know how much memory a db can hold of an individually device you do not get around the work to modify a plugin for all concerning platforms.
At the moment there are still only two global players on mobile market:
Android:
But there is a way to get max memory usage of a sqlite-db of android here
Have a look on the answer that states: Android's SQLiteDatabase class includes a method that sets a maximum database size, setMaximumSize(long numBytes)
I made a plugin for android recently and so I'm pretty confidend that making an android db-max-checker-plugin is a peace of cake. Of course you have to implement code that handles sqlite-db operations on your own but there are tons of tutorials out there.
And here is some kind of startup app for IOS, Android and WindowsPhone(Hello World).
IOS(Apple):
Unfortunate I've not found any reference that matches your request but
here is a good note that states that sqlite-db is limit only on hard-drive of your device and the programm's footprint(how much objects are allocated, imported resources like images etc.) is rather moderate. In terms of cordova allocating a great deal of objects is no need. Nowadays an IOS-device comes with a big hard-drive space so the chance that there is a memory-shortage sometime is less likely than when it comes to android.