First, what kind of computer do you have? Is it a desktop or laptop? I would really recommend that you get an internal burner if your computer is a desktop drive since there is much less overhead for a Serial ATA connection versus a USB connection since Serial ATA is a point-to-point connection so protocols for sharing a bus with multiple devices are not needed, and having the drive connected directly to the power supply unit instead of USB will guarantee better power availability. Also, since external burners generally violate the USB power limits (with the exception of USB-C, which does require a power limit high enough to power these drives), several computers whose motherboards are cost-reduced to the point that they can only deliver enough power to meet the USB specification will fail to work with these drives unless those drives are attached to a separate AC adaptor.
Second, Pioneer has released firmware updates for the BDR-209/S09 series (excluding the BDR-209DBK which was never intended or advertised to support BD-R QL media) and the BDR-211/S11 series to support your Sony 128GB discs. However, Pioneer really took its time to release those updates. You might just have to wait for Pioneer to finish developing those firmware updates for your drive.
Third, there are three organizations that currently make BDXL burners that I am aware of: Pioneer, Philips & Lite-On Digital Solutions Corporation (a joint venture between Philips and Lite-On also known as PLDS), and Hitachi-LG Data Storage (also known as HLDS). TSST (a now-former joint venture between Toshiba and Samsung that used to make these burners) is bankrupt. HLDS only makes external Blu-ray burners and quit making internal burners a long time ago. I recommend against TSST since its bankruptcy and probable liquidation will probably mean no more firmware updates. I will also recommend against PLDS (which sells its drives under the Lite-On brand) since it does not deliver timely firmware updates on its own DH-16AFSH-Premm premium internal DVD burner (unlike some rebrands of the same hardware by Vinpower Digital like the Optiarc AD-5290+ and the Plextor PX-891SAF PLUS). This leaves Pioneer and HLDS.
HLDS (which sells its drives under the LG brand, and a few notable rebrands like Asus) generally prefers to rely on generic burning strategies, and includes only a few specialized strategies in its drives for certain media. Teardowns performed by cdrinfo.pl show that LG drives generally are heavily cost-reduced and not so well-built. This makes sense because some people have written good reviews for drives, but others write reviews saying that the drives died after a while. However, the generic algorithms have been really good as far as generic algorithms go. However, no generic algorithm will work well with all media.
Pioneer generally prefers to rely on a large library of specialized burning strategies for different media. These drives will either fall back onto a generic strategy if there is a generic strategy and no specialized strategy for that type of media, or refuse to burn media if there is no specialized strategy and no generic strategy for that type of media. Your drive lacks both a specialized strategy for the Sony BD-R QLs that you bought and a generic strategy for BD-R QLs in general, so it has to refuse to burn the discs. This might be solved in the future if Pioneer releases a firmware to give it a strategy to burn your media.
You might also want to take a look at SONY Japan BD-R QL (128 GB) and Pioneer Writer BDXL QL (128 GB) compatibility to see some people running tests on LG and Pioneer drives on this media. It looks like the LG does a better burn than the Pioneer drive on those Sony discs in those tests.