It's likely not the modem. The reps on this community will probably want to run remote diagnostics on your HughesNet equipment, so it would be best to leave it on, if you can. They may also need some information from you to locate your account, but if they do they will ask for it via Private Message rather than in public.
By the way, though streaming can usually be done under normal circumstances, satellite internet isn't a replacement for satellite TV, as streaming uses data very quickly. Depending on your plan, you could exhaust your plan data after watching just a few shows or movies. Depending on the streaming service, it can be upwards of 3GB or more per hour in HD, and 700MB or so in SD. Granted, I have no idea how often you're at your cottage, but I thought it important to mention this.
With this said, streaming right now may be difficult, as the system load is inordinately high due to so many people being home as the result of the pandemic and because of a prioritization policy that was put in place on March 20th to help those working and schooling from home. During periods of high system load, which is nearly around the clock right now, things like streaming and file downloads may take a back seat to those working and schooling activities, which again means that streaming may not work very well.
The reps are on M-F from approximately 9AM to 6PM EST. Hopefully you'll get a reply from one tomorrow.
I have a similar setup at my property where I use HughesNet, and your electrician is correct, it shouldn't be the inverter, at least not in terms of capacity. The Modem/Router does not pull many watts, so your inverter should barely notice it. However, if you are powering up and down when you leave/arrive (I do that too) you should probably unplug your modem when doing either operation. Also, when your modem is plugged in, make sure you are using a quality surge protector and/or UPS.
That may not be at all related to the problems you are having, but it's a good rule of thumb.