After looking at several pictures of the Sirio 2016 loading coil online, it appears to be tapped at about the same point as the Sirio 827. It is an L-Network. From the coax connector's perspective, there is an inductor to ground and an inductor inline with the vertical element.
With the vertical already having a matching network, attempting to load the antenna at a different frequency will be more difficult than it would be without the built-in matching network since there is a fixed inductor to ground.
I think the only way to do it is to disconnect the loading coil from ground (which is inside that waterproofing coil cover) and to use a tuner. The top half of the loading coil would stay in series with the antenna, but the lower half from the tap point down would not be connected to anything.
This does mess up both the tuning for 11m and the DC grounding it had, but both problems could be solved with a tuner at the other end of the coax (although it depends on tuner style for the latter). If it is a typical high pass T-tuner, then a 10k resistor capable of handling the expected voltage could be used to return the antenna to DC ground for static discharge purposes.
With the coil modification, there will be a mismatch at the antenna connector, even at 11m, so this means the length of the coax will have an impact on the tuners ability to match a given frequency as the coax is acting as a frequency dependent impedance transformer. It should not affect the SWR, but the length will have a significant impact on the range of impedances seen by the indoor tuner.
That is not a bad thing though. Again, the length of coax is not changing the SWR, it is the complex impedance that changes as it rotates around a constant SWR circle, so, if you find that on a particular band the tuner cannot obtain a match, simply adding a piece of coax can move the impedance to something the tuner can handle. This is a case where a mismatch at the antenna is a benefit.
The other option is to create a new mounting bracket that can accommodate the ground radials and allow you to isolate the existing mount and unused radial hub from ground. You could then drive the antenna from the bottom end of the loading coil where the ground was (ignoring the existing connector and tap point) and run the entire loading coil in series with the vertical element.
