My employer has begun laying off employees. However, unlike nearly
everyone else who has been laid off, I'm being asked to work
throughout my 11 month severance period rather than just getting a
cheque for "garden leave" (i.e. I have to work during the notice
period, from home, for 11 months, rather than just getting a cheque
for 11 months of salary and being told to go away).
I think you are confused a bit. Getting a cheque in lieu of working the notice period is just that - giving people money without them needing to work the notice period. It may be done for many reasons, but most of the time it is done to cut costs as keeping employees working carries more costs than just their salaries.
Garden leave is still working out your notice period, but due to conflicting interests, for example, said employee planning to work with a competitor, the employee may be put on garden leave to restrict the number of trade secrets they can take with them. While in theory being on garden leave can mean you will be given alternative tasks, like say cleaning office premises, in practice the majority of the time it's being paid to NOT start a new job. With garden leave you will rarely be just given a lump sum (rare enough that I've never heard of that happening) but rather you will keep getting paid every month as before. Garden leave is something entirely done at employers' discretion.
The work I'm tasked with, surprisingly, is still important stuff, but
my heart just isn't in it. My suspicion is that my employer is simply
trying to get me to slip up so they can fire me with cause.
That sounds like a substantial stretch on your part. A notice period is a tool that works both ways, it protects both employees from being fired without financial backing, and it also protects the employers so key employees cannot just get up and quit the very next day. All they've asked of you is to now keep your end of the contract and work out the notice period you've agreed to in your contract, and nothing more. Nothing unfair about it, just like you are expecting to be paid for the 11 months of the notice period, the company has planned to get work out of you all the way through it.
To ease your mind a little, as you already are serving your notice period then to fire you with cause now, and to not become open to a lawsuit, is pretty darn hard. If that was their plan, it's very poorly thought out, and generally speaking, it will be a lot cheaper for them to just pay you off for the remainder of the notice period than to deal with the litigation of unfair dismissal (getting put on PIP while already serving notice would be a prime time comedy). That is as long as you will keep doing your best to deliver your work, there is no protection if you will start to intentionally screw up, or makeup lies to get out of doing your job.
Are there any actions I can take that would compel my employer to drop
this game and just pay me out like everyone else?
I won't go over your list as that's not needed, but I will highlight one thing - DO NOT LIE TO YOUR EMPLOYER. Being caught in a serious lie like this will very likely because enough to just fire you, without paying the remainder of your severance. And all those lies are very easy to found out as such, getting you in very hot water, taking any possibility of a reference together with the next 11 months guaranteed income.
I obviously don't want to be fired with cause
Good, then don't do it, just keep working your notice period and enjoy that you have a year of guaranteed pay while you can hunt for a new job, and have a guarantee of a stellar reference.
I'd like to make myself appear to be a potential problem/lawsuit in
the making without destroying my public image and long-term career.
You can't do both at the same time. You either are a problematic person who would try to weasel out of their end of a contract and is willing to do so in a dishonest way, or you are a reliable and honest person whose word is his bond. Up to you to figure out which one you want to be, but I can say without a doubt that the honest one is a much better long term strategy.
I also have to address one thing that you didn't ask, but should've, so adding a made-up question:
How can I discuss my worries with the employer so that we canfind a
better, mutually beneficial, solution that makes us both happy? As I
am very unhappy to work a job which ends in me being fired in a year,
and I'd rather just take a long vacation.
You can do so by having an honest 1-1 with your boss and explain your worries about being fired for cause, and motivational issues with him. Since they want to get a year out of work from you, they clearly value you as an employee (as I said before, it costs them more to keep you on than it would to just cut you a cheque, but they are also getting some work in return) and want you to keep performing. So have an open and honest dialog with your boss, and see what you can come up with that makes you both happy.