If he's contacted you, he's probably keen on having you to do the job.

So offer him "open book accounting". Show him how much you had put in for materials and labour (itemise them) and how much for contingency and how much for profit.

Then tell him you will do the job at the price he tells you.

I bet he gives you the job at very close to your original bid. I've done it twice and it worked both times.

I quoted for a job 3years ago to lay oak flooring, the customer had already bought it & if I remember correctly had paid around £3000 for it. I quoted £3200 to fit it. I received a phone call one day saying he had another quote for £1800 but he would like me to fit it if I could lower my price a little.
He went about it in a nice way & I too wanted the job but said,

"No, that's my price, the other guys price seems awfully cheap & if it's not done right he could ruin your flooring that you've spent £3000 on & you will have wasted a lot of money if it's a mess, have a think about it & you can get back to me, get another quote if you like & compare that"

So I left it at that, he called me back 10mins later to say go ahead, I presume he just had a chat with his wife about it. He said later he was glad he went with me as the other guy did not really inspire a lot of confidence & I appeared to know what I was talking about.
The way I look at it is if they are asking you to come closer to someone elses price, they must want you to do the job, otherwise why did they just not go for the lowest quote in the first place, it's all about selling yourself. I have no idea if the other guy was any better or worse than me.

I have since done other work for the guy, so he must have been happy. You hear so many horror stories out there about poor tradesmen, so assure them you are good at what you do & know your stuff. I just try & do a good job & provide a good service while making a good living at it.

You could always take the job at the lower price and try to make up the difference in "extras to contract". Perhaps not strictly ethical but it would seem the way things are ATM.
There's always going to be unforseen/uncosted works and the prices that contractors seem to charge for these "extras" can be disproportionate as they know they will get the work because they're already on site and ready etc and will get the work.
A LOT of large contractors are making all of their profit this way at present.
Not really a recommendation, more of a suggestion.

I know what you mean backhander. But it's just a domestic job, so no "contract" to speak of. Most jobs I do involve lots of little bits and pieces I never charge for, for the sake of goodwill. The shitty clients, however, always get hit 😈

Fair enough, and probably the best outlook.
Once you've been burnt by a contractor in this manner you try not to use them again.

I've made it to Sales & Marketing Director so I guess I must know what I'm doing!

sales and marketing titles are pants – they should always be separate roles as they have totally different skill sets IMHO

If its essentially labour you are selling, which it is then your other quote being much lower must mean its either: less time or a lower hourly rate. That simple. You should be saying – this is how long my estimate to lay it is, and in that time i will do X Y and Z which means the finished product will look like X and have these features. If you reckon it takes say 30 hours to lay it and the other dude can do it in 15 then it will be less than half as well finished.
We do most of our work on T&M based on a scope. We don't reduce rates and we do not negotiate. Never have and never will. We don't rip our clients off and we won't get into p1ssing competitions. We do good work consistently and have a back catalogue of very happy clients, we've never lost a client and on my watch we never will. We often get companies wanting a cheap job or getting 'bought' quotes. We never even try to match them all we do is point out what we would do, very open on financials. last week we had 2 come back who have had their jobs go absolutely t1ts up (one for the second time). The quote to sort the mess out is almost twice the original and we're not even loading it. One day I always hope they will learn…