The current AsciiMath input jax is really the v2 one pasted into MathJax v3 with some shims and glue functions, and it really needs to be made into a proper input jax for v3. It's on the to-do list, but other things have taken priority. It turns out that the display parameter doesn't actually work properly, as you have noticed.
Here is a configuration that will do what is needed:
loader: {load: ['input/asciimath', 'output/chtml']},
delimiters: [['``','``'], ['`','`']]
const {AsciiMath} = MathJax._.input.asciimath_ts;
Object.assign(AsciiMath.prototype, {
_compile: AsciiMath.prototype.compile,
compile(math, document) {
math.display = (math.start?.delim === '``');
const result = this._compile(math, document);
const mstyle = result.childNodes[0].childNodes.pop();
mstyle.childNodes.forEach(child => result.appendChild(child));
result.attributes.set('display', 'block');
MathJax.startup.defaultReady();
Here, we configure AsciiMath to use two different delimiters (the order matters), and hijack the AsciiMath compile() method to handle our needed display mode. We remove the top-level mstyle element that AsciiMath adds to set the displaystyle, and set the math element's display attribute to block for display mode (leaving it unset for in-line math, since that is the default). This makes displayed math be a displayed equation as well as typesetting in display style. If you want it to remain an in-line expression but using display style rules, change
result.attributes.set('display', 'block');
result.attributes.set('displaystyle', 'true');
instead.
I think this should do what you are looking for.
Davide