Ionic doesn't play any role here, as you probably know.
For Cordova I fear you will have to bundle a custom sqlite build specifically for spatialite, which means a custom plugin for Cordova is probably required.
I don't know iOS enough but for Android give a look at https://github.com/geopaparazzi/libjsqlite-spatialite-android.
Keep in mind that heavy processing should probably fit in a native code plugin. A mobile browser doesn't provide the same resources and performances of a desktop browser.
giovanni
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Sorry, I didn't mean a sqlite build but a sqlite library to intervace the plugin's native code and sqlite/spatialite...