Yes, you can operate the speaker unmounted. It will sound very weak when doing that due to sound pressure cancellation.
I thought of something during my first cup of coffee this morning. It was part of my KT training I had at work for root cause analysis. We can easily jump to conclusions of what the issue may be based on past experiences that may cloud our judgement to what the actual issue is. We already know the side effect that the amp cuts out when you are driving the speaker harder with heavy strumming vs non-aggressive playing. What would be helpful if you had a speaker coupling cable that had a female end to you can connect the speaker cable to another amp. Drive the speaker while it is still in the combo by another source to see if the sound cuts out under similar circumstances.
Where I am going with this: A combo amp is almost like having a built in HALT test but without the extreme cold temperatures associated with that type of test. More of a vibration table created by the speaker. The more voltage and current driving the speaker the higher the vibration from that moving part will transfer mechanical energy into the chassis. This will be vibrating power and preamp tubes, internal cables or wiring that is a press fit connection as well as soldered wires on the PCB. Any of this mechanical energy transfer could reveal a bad solder joint, weak connection, or bad part. Since the speaker replacement is a simple change and time it would take to determine if the speaker is the cause of the issue will be far less than having to analyze the actual preamp board inside the amp. Sort of like when a Roadster stops working, the common practice is to replace all of the JFETS without determining which one is at fault. The JFETS are typically used as solid state switches in the amp for some modes or voicings not managed by relay contacts, they are also commonly used in the strobe mute circuit. Replacing all of them is a form of shot-gun approach which does not really isolate the root cause of the issue. I will be optimistic that the issue is associated with the moving part, the speaker or its electrical connections and not something inside the amp.
If you have not pulled the chassis out of the Mark VII, it is a tight fit (it was that way with the head). I pulled the chassis out for S&Gs as I end up doing that with all of my amps when I first get them. Sometimes a good image of the amp before it gets used and abused may be helpful in determining a component value (resistor) in the event something fails down the road. Perhaps with an old-School amp that would be worth while, with the modern designs, not something that would be helpful. What you will find inside is this: It should be safe to remove the chassis from the amp without risk of electrocution from stored energy. Just keep your fingers out of the guts and all will be fine. Also, the chassis is where all of the weight is so expect it to be heavy. Probably best to remove the power tubes and set them aside for reassembly. Sure if you wanted to power up the amp and try that outside of the combo shell, it can be done but more than likely it will be inverted since there is nothing to rest the amp on with the power tubes installed. That is generally what I have done with the Mark V90, Roadster, or even the RA100 when tube rolling.
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Oh yeah, I just remembered something with the Mark VII when I first used the footswitch. I did not have the channel selector set to CH2. I found the amp was doing some weird things when I had it set to CH3. The JP2C also had CH2 marked for footswitch use but that amp never did anything funky if the 3-way mini toggle was set on a different channel. Getting the chassis back into the combo shell will be the harder thing to do. I recommend connecting the power cable first before you slide the chassis back into position. I found it difficult to try to connect it after the fact but that was with the head version and very little room for your hands to fit in that tight space. Combo may be easier.