I had looked into this, but didn't find anything either way. After another search, I (finally) found a technical manual for the PC, and of course the slot is limited to 35W. Damn. Suppose I'll have to rig up something with a powered riser, or go for a GTX 1030 (not a fan of the latter). I found this cheap riser on AliExpress, it ought to work (I just need a SATA-to-Molex adapter to power it). Of course I'll have to mod the case some to fit the GPU somewhere else, but that ought to be doable.
Still, the number of variables in play suddenly shot up quite a bit. If I need a riser anyhow, there's no need for the GPU to be low profile (other than fitting more easily), so why not look at an ITX RX 570? The price is the same as a 1050Ti, after all, and with some downclocking and undervolting it ought to be possible to run it off a 240W power supply, even an old one, while dramatically outperforming the 1050. And RX 570s can be found very cheap used. But to fit a bigger GPU, I'd need to replace the CPU cooler - a Cryorig H7 ought to work fine, and is very cheap, but it's another step before finishing this.
I was prepared for this not going entirely smoothly (if it was that easy, everyone would be doing it!), but this threw a spanner in the works. Anyhow, thanks for making me look into this once again, it would really have sucked to buy a GPU and have it not work.
Edit: One question - would I still need a powered riser if the GPU is powered separately? I know powered GPUs still draw power from the PCIe slot, but this ought to be somewhat self-regulating, no?