[David R. Falzarano] “In a situation like this, what should I do? A lapel mic would be a good idea? A more directional and less noisy mic?”
The first thing one needs for good dialog captures is a quiet environment. You don’t have that. First thing you should do is set expectations by telling your client that making a recording in a noisy environment is going to result in a noisy recording. You can’t change the laws of physics.
You can try a lav., but it’s only going to help so much. It will not eliminate the background. And resist the temptation (I’m not kidding here) to use a directional lav. as these introduce their own problems (head movement taking the voice out of pattern, proximity effect becomes a variable, etc.). Even professional actors and newscasters have trouble with directional lavs, and they’ve had the training and have the skills.
You can try a shotgun, but a shotgun(interference tube) mic near reflective surfaces (walls, ceilings) causes it’s own audible artifacts. And even when perfectly placed, a shotgun will only lessen a noisy background, not eliminate it.
Your best shot is perhaps an earset mic. Since it’s worn on the head, it turns with the head, so you can use a directional mic without much of a problem. Since the mic capsule sits near the corner of the mouth, it’s about as close as you can reliably get, so your signal-to-noise ratio will be as high as you can get it. Will this eliminate all the background noise? Of course not. All it can do is minimize it. But it may well minimize it to the point of practical inaudibility. Broadway shows use a fair amount of earset mics for example.