With what payload?

>The comparatively lightweight rocket amounted to 330 pounds

That's a total weight less than any warhead available for ATACMS: https://en.wikipedia.org/wiki/ATACMS#Comparison

And then there are the engineering challenges of flying very specific trajectories capable of single digit meters of CEP at those ranges, surviving a ballistic reentry to ground rather than parachuting, and being robust enough to survive being bounced around in a sealed canister for years at a time with no maintenance and then successfully launching in anger at short notice.

Or we could avoid weaponizing kids and let them explore space for the sake of exploration. Give them a no strings attached grant and let them decide later on what they want to use acquired experience for.

The entire rocket is smaller than the (smallest) warhead of an ATACMS; for a missile, you need to be able to deliver a useful payload at range, range without that is meaningless.

(Also, DoD has plenty of missiles with a longer range than the ATACMS.)

Why this obsession to get the world sleep walk into world war3?

They're not expensive or well-connected enough to get DoD work. Would have to promise to build different sections of the rocket in different swing states, etc.

Why DoD (US) instead of Ukraine Gov?

serious answer but without a dod contract and appropriate licensing, putting any kind of guidance system other than "keep it pointed straight up" on a rocket like this definitely makes it fall within a few things:

ITAR (international traffic in arms regulations)

EAR (export administration regulations)

CCL (commerce control list)

BIS (bureau of industry and security)

MTCR (missile technology control regime)

Just to start... Let's just say if you want to guide something like this you'd better be sure that the federal government absolutely knows who you are and what you're building, and for what purpose, and that everyone is a US citizen who is eligible to get a security clearance.

I guess I didn't even get into the other explosives/munitions related licensing and certifications and storage/preparation facility requirements for very large solid rocket motors (ATF)

Ukraine does have "Smerch" rockets - the engine part is about 500kg, 6m length, 0.3m diameter. In the dumb version it carries 280kg payload to 70km. The high-precision version carries 170kg to 120km. You can decrease payload further - to about 60kg - and hit the bridge (not that 60kg would destroy it, yet given the cheapness and availability of the rocket, one can use it to say regularly hit military cargo trains and trucks on the bridge while also damaging the rails/pavement for several hours in a row, and also one can use it to attack the S-400 defending the bridge, etc.)

I think what Ukraine hits, be it with own or Western weaponry, isn't just the matter of capability. I think they are trying to be very careful to not trigger nuclear response (my personal opinion is that this fear is overblown - of course i have this opinion while sitting comfortably in an armchair in CA). Also i think they are trying to not scare the Western partners as a country which, just to name a few, was the powerhouse of the USSR space and ICBM programs and has several working plutonium producing nuclear stations, and thus definitely has very interesting options of its own strategic defense.