Probably not worth the time to try swapping out the case etc, but you can get a pretty good low profile video card, such as this one from MSI
MSI GeForce GTX 1050 Ti DirectX 12 GTX 1050 TI 4GT LP 4GB 128-Bit GDDR5 PCI Express 3.0 x16 HDCP Ready Low Profile Video Card
https://www.newegg.com/Product/Product.aspx?Item=N82E16814137081&ignorebbr=1&nm_mc=AFC-C8Junction&cm_mmc=AFC-C8Junction-_-na-_-na-_-na&cm_sp=&AID=11877032&PID=1800524&SID=
And you can also get larger wattage slimline PSU's, like this one too.
ReplacePower 420 Watt Mini ITX / TFX Power Supply
http://www.outletpc.com/wr4972-replace-power-rp-tfx-420-tfx-12v-420w-psu.html?gclid=Cj0KEQjwxPbHBRCdxJLF3qen3dYBEiQAMRyxSxHWWxH3cjS1rqn4bxRuiwdoCMxvkLza9cgqcLH3UXoaAkhW8P8HAQ
You may need a 24 to 14 pin power adapter (not sure, but you can check the power connection to the board, and if its a standard 20 or 24-pin you are OK, but some Lenovo's have a 14-pin connection that requires an adapter like this one
24 Pin to 14 Pin PSU Main Power Supply ATX Adapter Cable for Lenovo IBM
https://www.amazon.com/Power-Supply-Adapter-Cable-Lenovo/dp/B00WE5KSFQ
That said, I cannot positively tell you that they both with fit your system, even though it appears that they should, so you will need to look at the dimensional specs for both the PSU and GPU and do some good measuring of your existing PSU etc.,
Your system appears to have a Low Profile, PCIe x16 (Version 2.0) slot with two punch out faceplates needed handle the double width/slot face plate of the GPU, but you need to check to be sure.
Note: A PCIe 16x Version 3.0 card will work in a version 2.0 slot without any problems or negligible performance hit.
ideacentre 300s-08IHH SFF Platform Specifications
http://psref.lenovo.com/syspool/Sys/PDF/Lenovo%20Desktops/ideacentre%20300S%20SFF/ideacentre%20300S.pdf
Your case also has a provision for an optional front system fan too, which I would recommend if you upgrade the card given the small form factor case ( if not already installed). It should take a standard fan size, and have a nomal system fan plug on the motherboard. See page 32-34 of the Hardware Maintenance Manual
Note that the System fan is not thermally controlled like the CPU fan, and thus runs full speed all the time, so look for a quiet one.
Good Luck,