How do you beat Tyl Regor (boss of Uranus)?

Radiation damage, as you've noted, has a large bonus vs his alloy armor, but leaves you with a damage penalty vs his shields, and provides no benefit vs his "cloned flesh" hp type. It would seem that viral or cold damage provides you with the best combination of strengths compared to weaknesses, but magnetic could also help destroy his shield if you felt it necessary (with a penalty toward his armor).

In this situation, it might be helpful to have your primary equipped with one element, and your secondary with another element, then switch between them as necessary. Lex Prime and Euphona Prime come to mind as secondary weapons that might be suited to this fight, but other options such as the Gammacor or maybe even the Seer could work

Tyl Regor gets around very quickly, as you've also noticed, which leaves a couple key strategies:

Have a weapon capable of dealing large burst damage very quickly.

Your Hek is a good start; make sure it has an Orokin Catalyst installed and high levels of mods such as Hell's Chamber, Point Blank, and elemental mods as desired. Remember: if you have unused mod capacity, empty mod slots, or all the mods are not maxed, the weapon is (almost always) not being used to its full ability. Scattered Justice is also very helpful.

Slowing him down.

I know you said ability usage is limited, but there are ways to get around this, such as:

• High efficiency builds on your Warframe

• Leaving some empty mod capacity on your Warframe so that you start with extra energy (contrary to weapons, unused mod capacity on a Warframe actually serves a purpose)

• Taking your time killing as many enemies as possible on the way to the boss room so that you have a large energy pool to work with by the time you reach Tyl Regor (The "Flow" mod helps with this).

• Energy Restores

• Zenurik (included for completion; by the time you unlock this you won't need it for Tyl Regor)

In light of that, there are 2 popular ways to slow down Tyl Regor: Frost, and Nova.

Frost

Build for efficiency and range. Place a few snowglobes in the arena. This serves the double purpose of both slowing him down so you can get a few shots off, and greatly helping your survivability.

Nova

Nova's fourth ability, Molecular Prime, will change the speed of all enemies around relative to Nova's power strength. High power strength means every enemy around will be slowed to a crawl. Nova is not the easiest frame to survive solo, however, so Frost would be my suggestion.

A Hek with Hell’s Chamber and a reasonably leveled point blank should be able to carry you through the whole star map. Throw in additional mods to taste: reload speed helps on Hek while you get used to it. Rad or corrosive are good against grineer.

Hek is also good enough to warrant a potato and some forma if you need the mod space. The direct upgrade Vaykor Hek requires MR 12, so that’s a ways away