Since the actual Roku people seem to be either unwilling or unable to work on this problem, lets see what we can do as just ordinary users.
There are a couple of hidden maintenance menus which provide access to information and control functions over the WiFi that are not normally known about nor available to the "average" user. One is the "Network" menu and the other is the "WiFi" menu. These may allow us to gather some information which may be useful to you (and others) about what is going on. From there, we might be able to develop a solution to your issue, or at least understand it better.
First, before we go mucking around in these hidden menus, understand that they provide access to things which aren't normally accessed by users. hence, it may be possible to muck things up worse than they are now, or possibly un-muck them. But, care in what you do when in them is very important.
What I am suggesting is as follows....First we will use the "Network" menu when you are connected and running to get a bunch of information about how things are running when things are running, and then, you can get that same information when things are malfunctioning. This will give us a possible view of what the problem is. If that doesn't do it, then we will go into the "WiFi" menu, where much more detailed technical information about the WiFi connection itself is available, as well as status of the actual WiFi hardware inside of your TV. It is also possible to force the WiFi hardware and software to reset and reload, which takes into a bit more dangerous territory. So, lets start with the "Network" menu, which is more benign.
To get to the "Network" menu, you must put the TV into the Home screen. Then using your remote, you will enter a button press sequence of 10 button presses. These must be done briskly and without an interruption once you start, or it will not take you to the hidden menu. The sequence is:
Home, home, home, home, home, right, left, right, left, right
(that's 5 home presses in a row, followed by right, left, right, left, right)
If you are game for working with this a bit, see if you can bring up the "Network" menu and copy down the information it provides when things are normal, and then when things are abnormal.
(By the way, as a side note to the Roku personnel reading this post, THIS is what user support on an on-line forum looks like)