[RPG] How to handle modern-day weapons in D&D 5e

General advice for any issue with a trouble player is to talk with them and clearly explain the trouble that you are facing. You should definitely explain the problems you are having with trying to keep track of all the custom stuff and see if the two of you can find a solution. However since this is a question specifically about Savage Worlds, I figured I'd address your specific concerns in relation to the system.

You mentioned that you are both a new GM and new to Savage Worlds. One thing that the creators of Savage Worlds often say is that although it's tempting to play with customizations right off the bat, you should really try to play "by the rules" first. There's a great section in the GM section about Edges and Hindrances that encapsulates this pretty well:

For the most part, we’ve found you really want to keep the selection to less than a dozen powers, and half that number of new Edges or Hindrances. A lot of new Game Masters go crazy creating scores of each, but at the end of the day find that most of the stuff people actually take is already covered in the main Savage Worlds rule book. That’s not to say you shouldn’t have some cool new powers, Edges, or Hindrances—just that you should think them through very carefully, and add them mostly for flavor or to cover some very unique feature of the setting that the rules don’t currently cover. —Savage Worlds Deluxe, Page 128

Replace "Edges and Hindrances" with "Gear" in that sentence and I think the same advice applies. It's tempting to add all sorts of new gear, but at the end of the day you're probably just as well taking the stuff from either the core book or a published setting book for the setting you are playing in.

You didn't say what setting you were playing in, but since you listed some modern weapons, I'm going to guess you're playing in that time period. In that case, there shouldn't be any real reason to create custom weapons because there are so many in the core rules. But notice that there are very few "unique" weapons", especially in the modern era category. All of the guns have roughly the same damage, rate of fire, shots, and special notes. For instance, all assault rifles deal either 2d8 or 2d8+1 damage, have a rate of fire of 3, have 20 or 30 shots, and have AP 2, auto-fire, and possibly 3-round-burst. If you want a new assualt rifle, all you really have to do is pick one of the ones on the table or mix a few of their stats and slap a new name on it, then pick a cost and weight for it.

I've worked on some third party Savage Worlds products and in all honesty this is exactly what most Savage Worlds writers do for anything that isn't groundbreakingly different (and if it is, we usually just add an extended notes section explaining what it does). We do this because we know the numbers already work and because we want to match player expectations. Say that I was creating a setting that had an LSAT rifle (a potential replacement for the M16 used in the US Army today). I'd just pick the M-16 stats and change the weight to be lighter. That's it; there's no real reason to customize further. You seemed to have the same idea for this with your bangstick you mentioned at the end. Just have him buy a bangstick for a price that seems reasonable for the setting, then change the weight and any essential trappings and you're good.

It sounds to me like there might be a bigger problem too in that your player is trying so many different things. Shields, shotguns, pistols, grenades, knives, bangsticks, and dogs. Again, I don't know what your setting is, but it might be appropriate to start enforcing consequences with all that. How is he hauling all this stuff around (if you want to use the encumbrance rules, what penalties is he taking)? Does he freak out every person in town when he comes armed to the teeth with these weapons? Do people not take him seriously when he has a pistol and shield? How much in debt is he to pay for all these different weapons each with their own ammo? Where the heck did he find a shield in a time when people are toting around guns?

You did note that he was starting to be spread out pretty thinly, so you might also want to encourage the player to start specializing in a more limited set of gear and stick with it, saving his other ideas for a different character. Moreso than in Dungeons & Dragons, there's not much point to continually upgrading gear since those in the same category have basically the same stats and there are tradeoffs between different types. Also depending on the setting it may not make sense to switch weapons (if you're playing a military game, you're probably staying with whatever you were assigned by the higher ups).

Also one last thing: you said that the character had a shield and guns. Having a shield is pretty much worthless in a modern game because all guns have armor piercing. He can carry around a Medium Shield for an extra 12 weight to get +1 Parry and +2 Armor vs. ranged attacks, but it's unlikely that he'll get into melee combat (why bring a knife to a gun fight?) and all rifles have Armor Piercing 2, which will go straight through it as if it weren't there. It would provide 1 point of Armor against pistols, which generally have Armor Piercing 1, but is it really worth carrying a shield around, not to mention tying up a hand that would prevent you from using a shotgun or rifle, just for that? There's a reason that soldiers historically quit carrying around shields once gunpowder weapons were invented.

Yes, you are overlooking some major flaws

I'm only going to address the Major Flaws you're concerned with, since the rest is subjective to both your game play and how people play the class. This should help you get through at least a couple of misconceptions you have.

1. Pact Weapon (PHB. pg 108) - You can only have one of these, and due to that, Thirsting Blade can only be used with your one pact weapon. If your DM does happen to allow you to take two pact weapons, that's cool, but is clearly a house rule. Make sure your DM knows about the wording of Thirsting Blade with respect to Pact Weapons.
2. You have only one bonus action possible per turn (PHB. pg 189). If you have many abilities that use the bonus action (Cunning action, offhand attack, specific spells, etc) you have to choose which ONE to use.
3. Survival wise - largely dependent on magical item availability, your feats, and potion use. If you take fighter at level 1, you'll have higher hit points to start, but your saves will leave you mentally vulnerable. That's why I recommend Archfey pact below, to remove that problem.

The following part is only my opinion on the build itself, and where I would make changes to better fit your stated style of play.

Personal build recommendations:

1. Take 3 levels of fighter to become an Eldritch Knight so you can have weapon bond (two weapons bonded, functions very similar to Weapon Pact, allowing for 3 summonable weapons.)
2. Instead of Fiend, use Archfey as patron because the up close abilities and immunity to charm are superior for an up close fighter. Especially since Misty Escape gives you a free exit in overwhelming situations.
3. Feats - Heavy Armor Mastery, Tough.