Here's the math. The super cheapo HKG-TPE fares are about ~$200 USD in CX's pocket. Say you do that 10x annually = $2,000 revenue to CX.

Now, say you hit up the lounge each time you're in HKG and TPE. I think the lounge is about $15-$25 USD in HKG and not sure about TPE, but let's assume less maybe $10 USD/person. In essence, CX is going to incur $200-$500 USD on lounge costs for such a frequent traveler annually who is spending about $2000. Basically they're spending 10-25% of the revenue you give them back on your lounge fees in this case, since you have a lot of frequency but cheap fares. That's in addition to increased baggage allowance, greater staff/pax ratio at check-in, etc. It's not a good trade-off.

Maybe what you can do is offer to pay the difference to your employer. I'm not sure what fares you're taking, but there's a chance you'd only be out of pocket $50-$100 USD more per flight. Once you do the math though even that doesn't look so great because you're basically paying that much to get in the lounge. Which obviously doesn't make a lot of sense.

I think the bottom line is get an employer who doesn't pay for the cheapo fares. Somewhat surprising to me, since you cannot change those fares at all and they are non-refundable. One trip where you get sick, or the trip gets cancelled, etc and much of their nickel and diming will be for naught.