Garmin has intentionally crippled their custom map format out of concerns that it could impact sales of their own products. Unless you are using a Garmin device, there is absolutely no reason to use "custom maps". But if you want to stick to that format, the largest screen Garmin devices available are the Zumo 595 and GPSMap 276cx, both have 5" screens
https://buy.garmin.com/en-US/US/p/536397#specshttps://buy.garmin.com/en-US/US/p/539722I don't know anything about the Zumo, except that it's expensive. The 276cx has been very controversial and Garmin has alienated many long term customers due to bugs and poor performance. See this lengthy thread for example:
http://advrider.com/index.php?threads/garmin-announces-the-new-gpsmap-276cx.1178312/Regarding the limitations of the custom map format, it will only support a maximum of 600 map tiles at 1024x1024 pixels each. That is a tiny footprint and seriously limits the coverage area, especially if you want to work with high resolution imagery (such as 1 foot per pixel).
It also does not support multiple zoom levels and forces you to use jpeg files which can be inferior to tiff or png files. There are many other raster imagery formats that work across a broad variety of devices including Android and iOS apps. This gives you access to multiple zoom levels, where a different map is displayed with zoom level, allowing things like text to be readable as you zoom out. And there are no practical limits to how large these maps can be as the imagery is stored in a database.
You can get very nice large screen Android tablets for under $200 and if you want better accuracy you can use a bluetooth GPS such as the $100 Garmin GLO or even expensive receivers that can give sub-meter accuracy.
So, IMO, you should not waste any money pursuing a large screen device for Garmin Custom Maps, it will be expensive and very limited. If your Oregon doesn't meet your needs, perhaps a Montana would be good enough (4" screen)? You can get a factory refurb Montana 600 with full warranty for $300.
Otherwise, you should look at either an Android tablet or iPad with an app (there are many inexpensive ones) and possibly an external bluetooth GPS receiver. You can still use QGIS to create your maps, but you would just need to export them in a compatible format.