D&D 5E 5e Build Help: L15 Street Boxer

Long post, so here's the TL;DR: Help me build a Level 15 half-elf street boxer without using the Pugilist class. I know similar topics have been discussed before, but I'm looking for help with my specific situation.

RL background: A friend and I are building a pair of Level 15 characters for a friend's high-level game that we, schedules permitting, hope to play in. We have decided to make our characters brothers, one a half-orc and the other a half-elf. My friend, the half-orc, has designed a Valor Bard. I'm wanting to make the half-elf a scrappy street brawler.
The obvious solution for me is to use the Pugilist class, but I haven't had a chance to ask the DM if he would allow this. If he does not, I want to have an alternate build ready to go.

Here's what I'm working with and what I'm looking for: Rolled stats of 13, 14, 15, 16, 16, and 18. However, I would be skeptical of these rolls as a DM, so I'm going to offer to resolve in the DM's presence [edit: or do point buy]. I'm hoping I can keep these stats, though. I want to focus on striking over grappling. I don't want magic, but not-strictly-speaking-magical things like Ki are okay, however some of the higher level mystical aspects of the Monk don't really fit with the concept. Unarmored defense would be better, but light armor is acceptable. Unarmed attacks and knuckledusters (one or two) as my weapon of choice. I'll have to talk to the DM about the knuckledusters, but I'm sure I'll be able to refluff clubs, however, since I picture this guy as a Dex-based boxer I'm going to try to talk him into letting me add the finesse tag to them. Though less than ideal, I suppose I could be convinced to use a short staff/fighting stick (basically a club).
Classes: As far as classes go the best fits as I see it are Barbarian (Berserker or maybe Bear Totem), Fighter (Battle Master), Monk (Open Hand or Long Death), or Rogue (Assassin or Swashbuckler). The last in particular if I can sweet-talk my way into finesse on knuckledusters and/or Monk unarmed attacks so I can Sneak Attack ("Sucker Punch") with them.

If going with a single class, Fighter or Monk seem to be the best bet, though like I said, some of the more mystical aspects of the Monk don't seem to fit (many can be reflavored, but it gets tricky with some). Being a Monk/Fighter would work well at 4/11, 5/10, or 6/9, and I'm not sure which would be best. If I can get access to Sneak Attack, a three level dip into Rogue for Swashbuckler and the second extra d6 of damage. One or two levels of Barbarian (but probably not three or more unless I go pure Barb) would be nice for Rage, Reckless Attack, and Danger Sense (and Unarmored Defense if I want it based off of Con rather than Wis).

As crazy as taking three or even four classes normally would be, I like the flavor of it. He has little formal training, just finding his own way in life and picking up little bits here and there of whatever works for him. He wouldn't be as finely honed and disciplined as someone who devoted themselves to a specific field of training. Also, he has spent most of his life fighting alongside his brother, so has probably grown used to a Bard having his back.

Abilities: Str is good, but Dex is might be better. Con is always good. Wis I'm torn on. If Unarmored Defense from Monk, obviously I'd want it high. I see him as street smart and being able to read an opponent. On the other hand, he has more swagger than sense and often takes on the biggest, meanest opponent. This is the guy who would try to punch a dragon to death. Int I can go either way. Cha I can go either way, except Intimidation would be good to have high. With the stats I rolled, and especially with enough levels in Rogue or Fighter in the right place, I obviously have room to play with Feats. Mobile would be great, especially if I don't go Swashbuckler. Tavern Brawler might be worth it if I don't take any levels in Monk. There's a few more that might be good.

Fighting Style (if I take Fighter): Dueling if I only use one knuckleduster sounds good. Two-Weapon Fighting if I use two and don't have any levels in Monk also sounds good. So does Tunnel Fighter if (that's a big "if") it's allowed.
I have the flavor of the character. What I'm not good at is running the numbers in my head. So I'm coming to you guys for help. He doesn't need to be optimal, but I don't want to inadvertently cripple him. What would you recommend?

Based on what you wrote, I'm picturing your character being someone like Mickey from Snatch. In that regard, I think the character would REALLY benefit from focusing on the Swashbuckler Rogue. It captures a lot of the feeling of that street fighter nature, especially with being able to Sneak Attack without needing Advantage so long as you're in 1v1. Plus things like Uncanny Dodge and Evasion will let you avoid a good amount of damage.

For the unarmed boxing portion...if your DM won't let you use Knuckledusters or Gauntlets or whatever, then you need to hit that first ASI to get Tavern Brawler. The only other option is to pick up a level of Monk, but if you do I would push it until you get the Flurry of Blows. You might even be better off taking Monk at the start, provided your DM would allow you to be a Monk without actually "being" a Monk.

For other multi-class options, I think Barbarian is a great dip to get some Rages. But I also think that Fighter would be a waste for this character concept. I wouldn't go far into Barbarian, and its hard to say when to go get it. But starting off as a Barbarian could work if your DM does allow that Knuckleduster idea. It would fit well with the untrained brawler-of-the-streets thing.

If you do multi-class then I would decide quickly because whether Barbarian or Monk, you'll want to align your points appropriately for the unarmored defense. Based on what you mentioned, I think the Barbarian is a better choice since he will be tough...and a wise man does not punch a dragon. =)

thanks

Monk is a must, it is the "fight with your fists" class. I'd recommend the Way of the Long Death from the Sword Coast Adventurer's Guide.
Multiclassing is plausible, depending on how you want to be. Barbarian as your starting class allows you Unarmored Defense (Con), Swashbuckler Rogue gives you even more manoeuvrability and slipperiness, but could be replaced by the Mobile feat.

Overall I'd definitely recommend being a Barbarian 1/Long Death Monk 14. You'd need 13 Str, Dex, Con and Wis, but your main stats would be Dex (for attacks and AC) and Con (for HP and AC).