I think you're on the right track.

The customer needs to be told, up front, on the phone, your fee to respond. That fee ought to cover a minimal amount of time; say, one hour of troubleshooting / diagnosing.

As soon as you have a diagnosis, or you have looked at the proposed job, it's time to present them with a written specification of what you propose to do, and for how much. Have them sign it. Or not. Pressure tactics (Lady, your house is going to catch fire!) ought to be avoided- period.
If they defer - for whatever reason - treat them graciously, with respect .... the attitude you wish to convey is 'of course you want to act responsibly, and I'll be here when you want me.'
After all, you've been paid for the first part - the service call. Be like Sears, and get that credit card # before leaving your shop.

Once you have their approval, agree on the work schedule. Does it need to be done NOW ... or can a patch be made, with a return trip after the parts houses are open? Are other trades needed?

In any event, your appearance, demeanor, attitude, ... everything ... ought to convey the message that you're happy to be there, that of course you're worth your fees, etc. Duct tape and baling wire just won't cut it with a $1000 job!

If you present things properly, "negotiation" won't enter the picture. If it does, it should be on your terms ... as, say, a deduction if they remove the bushes that are in the way. Otherwise ... here's your price, period. No more. No less. And payment is on completion .... meaning NOW ... not next week, or when the property sells, etc.

Keep it simple, keep it pleasant. Never surprise the customer with a whopper of a bill after the work is done. Never argue, lose your temper, or respond in kind ... even if the guy openly asserts you're a crook, and don't know how to run a business, if changing that fan belt costs $160.

It can get hard if he asks, at the outset, "why so much?" Avoid this minefield. He's not interested in a breakdown of your expenses, your overhead, or your investment. He's trying to open Pandora's box. Just say "this is my proposal ... I will do this job for $X .... I can come back later, if you want some time to consider my offer."

It may sound callous, but remember: just because it's an emergency for them does not make it an emergency for you. Let them sell themselves on the job.

Chances are, the 'difficult' customers have already asked several of your competitors over, and have applied the same tactics to them. They're trying to play one against the other. They may be concealing information, such as prior problems or work, from you. If they have time to do that, then it isn't a real emergency for them, either.

Since getting my own license, many have asked me - as soon as we meet, without ANY discussion of what work is involved - what my hourly rate is. There is NOTHING good that can come from answering that question. So far, I have responded by telling them to show me a specific job, and I'll give them a specific price. None of these 'tycoons' have accepted my offer of this free estimate/ quote. That suggests that their motives are not pure.
But I digress ...