Sounds to me like you have a CIA chip problem. There are 2 of them, both surface mounted. Plugging devices into the Mouse & or joystick ports is NOT good practise when your Amiga is running. Same goes for Keyboard, Paralell/serial ports. Just don't do it when the Miggy running. Try another known working mouse to eliminate your mouse being faulty.
I've also seen boards with battery leakage just trash the entire board as far back as the Sound chips & rear ports. This damage isn't always evident untill you spray the board with an Alcohol based cleaning agent. What sometimes appears to be a small amount of leakage around the battery area can be more sinister than what you can actually see. Brake Cleaner for cars will work, but better to use any evapourating alcohol based Electrical cleaner. For spraying & hanging the board to dry, the "Fur" & discolouration to the PCB just shows the consequences of battery leakage.
Assuming you have no battery damage to the board & to resolve these symptoms you are experiencing, I'd do the following.
1) Try another known working mouse. If Right mouse button has the same symtoms, move on.
2) Ground yourself correctly & remove the board from the Case. Lay the board on a solid surface with some anti-static material under the board. Look very very closely at all the Capacitors & SMD Resistors on the board. Ruptured Capacitor tops & or a sweaty "morning-dew" look means they are fux0rd & need replacing. Look for this sweaty look on any of the boards components & be sure you have a good clean & bright light over the board while you do this. If none of this is evident, try pressing down firmly but gently all the socketed Chips to reseat, then retry the miggy.
3) At this point, if the symtoms are still there, you need to get hold of JJB @ Amiga France & he should be able to repair it for you. As far as I am aware, he replaces the surface mounted CIA's with socketed items, thereby making it easier for home users to pull the CIA's & replace them very easily using a PLCC Puller. I have managed to lever socketed Chips without a puller, but it's not for the faint hearted. You have a very very high risk of splitting the socket when you dont use a PLCC Chip removal tool. I only mention this because by pressing on the chips, you may expose another problem & end up with a Green screen when firing the Miggy up again. One reason for doing away with sockets in the first place, but repair costs for surface mounted stuff is high & very specialised.
Best of luck with your diagnostics.
Kin :-)
*Edit I can't emphasise the Importance of you being Earthed whilst doing investigation work like this. 1 nanosecond & you could have more problems than you originally started with! You mention use of an ocsilloscope above so I'm guessing you already know this, but assumption is the Mother of All F**k-Up's! ;-)