Kinetic Cannon
Kinetic Cannon fits very well on many different boards, as it ultimately aligns with Dooley's most common strategy, spamming Small items. Kinetic Cannon rewards you for sticking with the standard game plan, making it a consistent choice.
However, its scaling does tend to fall off later on, so aim to win by Day 10-12 or else you may need to pivot. Upgrading the Kinetic Cannon is very important to ensure it keeps up until that point, so be sure to prioritize vendors that could have it.
What's important is being able to Charge the Core as fast as possible, which in turn also scales Kinetic Cannon. You don't need your Small items to be super impactful since the Core and Kinetic Cannon should do most of the heavy lifting.
A variety of Cores can be used alongside Kinetic Cannon, but the two I recommend most are 
Weaponized Core and 
The Core.
Among the others, 
Ignition Core, 
Armored Core, and 
Critical Core are fine, but offer little in comparison. Since speed is a priority, the Core should generally be placed as the rightmost item (or close to), meaning you aren't benefiting from any extra Burn/Shield/Crit scaling. 
Companion Core obviously doesn't fit here.
Your Core of choice actually changes a lot and makes you have to prioritize different things during your run. Let's look at the two priority Cores and compare their strengths and weaknesses.
Weaponized Core + The Core
Weaponized Core is one of the most consistent Cores thanks to its efficient Damage scaling. Since Weapons are one of the most common item types available, you can actually utilize its rightmost scaling more consistently (at least early on). Dooley also has plenty of fast Weapons that take full advantage of this.
In many cases Weaponized Core can even function as the main Damage source on its own, provided it's being Charged fast enough. It's certainly enough to carry you through the early to mid game, which is what makes it a top choice for consistency.
The Core on the other hand enables a very different approach. When placed on the left side it can become a board-Charging machine, working perfectly with Kinetic Cannon since it will still receive Charges despite its leftmost positioning.
However, this doesn't really take off until you've got some upgrades to lower The Core's Cooldown. Plus, by not selecting a different Core you miss out on the powerful scaling that carries the early game. It's definitely the fastest option for late game, but it does come with the cost of early game power.
Kripp had a Restorative enchant on his Weaponized Core which really helps the build survive long enough for Kinetic Cannon to go off more than once.
Uzi
Uzi is a Dooley staple thanks to its incredibly fast speed. It's well utilized for its two second Cooldown alongside most Cores, but with Weaponized Core you can pump it's Damage like crazy. Always make it a priority to upgrade Uzi to Silver ASAP, as the extra Ammo is very important.
Kripp's Uzi was also Restorative, which definitely made it worth keeping to the right of the Weaponized Core. However, oftentimes in the late game you're better off placing most items to the left of the Core, as fights are too fast to make use of the scaling.
Schematics
Schematics is an incredibly valuable Economy item for Dooley. Its ability to generate items of any size can really help fuel your run, while also finding strong items for your board. It's not uncommon to pickup powerful replacements, or even high roll a strong duplicate.
Always remember to keep your Stash space in mind prior to leveling-up. If you're simply looking for gold, keep at least three slots open so you have a chance at a Large item. Oftentimes you may want to occupy all but one space, guaranteeing a Small item spawns, in hopes of getting a good item for your board.
