A false-shoulder is the best option, and how to accomplish it need not be a stumbling block, just figure out the work-around...
Proper annealing is the key.
A 308 expander will work and not cause neck fragility or splits in a properly annealed neck. Try a couple and see for yourself. Use adequate lube.
Brass splits when it is work-hardened but annealed brass is malleable, and brass stretches without fracturing when in an annealed state...you should see how the ammo factories mash it around with 100 ton presses. There may be a dozen or more steps to the overall process, and with a few annealings along the way.
If it were me, I'd first anneal the necks for uniformity, then form a few. (Annealing needs be done correctly... you need to have a handle on that process as it is an acquired skill; it's all too easy to over do it and burn the brass).
Having the proper tools at hand is always an advantage that one should always take advantage of...
A 308 expander is 2x diameter larger than necessary, but it only takes about 10 minutes to turn a spare steel 308 expander button down half-way to split the distance, or to whatever diameter you want it to be, and tapered for the task.
Eventually acquiring the right tools is a good idea. I'm moving stuff around right now, or I'd send you one, but RCBS might make one for you...they used to do this kind of thing, but it doesn't cost anything to call them up and ask. They have always been very helpful folks. RCBS or Redding may have expander buttons available in custom sizes, look at their catalog.
The jamming method will work, with mixed results, but you have to jam the bullet hard and the rounds will not be the same as normal. The result is a waste of bullets and powder and primers and my time, so usually I would prefer the mechanical solution, but both methods are workable.
A false shoulder load looks odd, but can function more-or-less as a normal round if you can fit the same quantity of powder in the case.
It's no harm to try a few cases just as they are with your 308 button and see how it works out. If the necks tend to split after just a few firings, it's due to the added hardness caused by work-hardening due to the extra forming steps and not necessarily by the brass stretching.
I'm usually looking to anneal after half a dozen firings, so you might expect to lose half of that due to the case forming process without extra attention to the state of the brass hardness.
