Yes, we should be able to point to a few things to help as you do fall into the right range to get the better wiring schematics on pre 2010 models.
Is this RV handy so that some testing can be done with reasonable time and effort? Or is it stored away from you?
Point being that we can cut the chase a bit by trying it and seeing what happens when the engine is running. The engine starting is very much like the normal car/truck making it less complex and more familiar than when we look at the whole RV.
But it also connects the start battery system and the engine alternator together with the coach 12VDC system. So if the relay and inside lights, vents, etc come alive, it can tell us a few things work and we don't need to look at those but can skip down a few lines in the chase!
So I kind of spend my day chasing things on electrical drawings and I've looked at the Winnebago set enough to feel comfortable with them. I will send along the link to them, so you have a choice of looking at them online or try to just use the snips of them I can post.
Different folks like different ways so you may want to get the complete picture or if that is not something you do much, the "snip" can be better!
You can often get a better look at these snips by clicking them if on a computer!
This is the drawing for your RV 12volt for the coach:
https://www.winnebago.com/Files/Files/Winnebago/Resources/Diagram/2008/08_331c_wire_162270.pdf
Page one is the part you need for the relay and has some important stuff to help understand what goes on in this critical area. For future maybe?
You have the right idea that there is a magnet gizmo on this as it is a latching relay. I've marked the control lines from the switch to the relay as easier to test at the relay then dig the switch out, perhaps? Or is it back out of reach on your RV?
There should be labels on the wire sides

as this example. We can "decode" the label here:
https://www.winnebago.com/Files/Files/Winnebago/Resources/Diagram/electrical_guide.pdf
The positive battery comes in on a wire marked LJ and goes back to ground on either LG or LH to connect/engage (C?) or disconnect/disengage (D?).
So are you getting the 12VDC on the left side of the relay? They do a good job of making the drawinglook like what we actually see! If you get it there, does it get connected to the right side when you toggle the momentarty switch to turn on or off? Does the big red wire on left connect to the big red wire on right and onto all most all the coach ?
I marked the control wires from the switch to the realy in blue and you should find 12VDC on LJ going into the coil of the relay and meeting ground on either LG or LH, depending on which way you push the switch.
One way to test if you find the realy not working is to pull LG and lh off and see if the 12 volt is getting to the relay coil as well as getting thorugh at the posts where you pulled the wires off?? Dirty connections, open coil wire? That's where testing will help sort it.
But there is a good chance it is something less complex like dirty battery cables along the route and you may not find battery getting to the relay??
At the left is the mode solenoid which does the connection between the start battery and coach battery by using wire LR bring battery from the boost/aux switch when pushed or the ignition run circuit when the engine runs. This battery goes through the solenoid coil and down to ground on wire FM that is often on the mounting screw. This is a big one that can fail often as it does lots of arcing and really often (every time we start the engine).
See if that makes sense with what you find with a bit of looking and testing?