I don't think you could get an engine capable of getting you hundreds of miles at that speed and altitude, much less the sensors and guidance system.
That said, you could probably make a glide bomb for pretty cheap. The two examples I gave above are in the hundreds of thousands, but I bet if you sacrificed some of the accuracy and payload, and you were really optimizing for cost, you could get that into the tens of thousands
It will be burnt up by the time it runs out of fuel, though.
For a good one it has to be hydro-formed out of seamless pipe, which is likely hot-stretched (like wire drawing, but using an induction heating coil in place of the die, and independently controlling the feed-rate and the pull-rate) in advance to retain consistent wall thickness in the engine despite being cross-section after hydro-forming. Also probably incremental hydro-forming with grain-structure-fixing re-heating between stages.
The hard part is just that most of the work is in making the hydro-forming dies/tools, not then using them to cheaply produce more engines.
Of course, you can definitely make a BALLISTIC missile with a couple hundred mile range using a solid rocket engine, and probably for fairly cheap (especially if you don't particularly care about accuracy)
Also: Scud missiles are comparable, made outside western cost inflation, and still ~$1M each. So I'm guessing you can't get a ballistic missile anywhere near $10k (at least, not one that isn't more likely to kill its owner).
https://disclose.ngo/en/article/war-in-ukraine-how-france-de...
(An order of magnitude larger helicarrier contract was cancelled eventually.)
0: https://www.theguardian.com/world/2003/jun/04/terrorism.davi...
This sounds similar to the Nth Country Experiment, conducted by the US in the 1960s. The idea was to see if three freshly minted physics PhDs could design a working nuclear weapon using only publicly available information. In less than 3 years, the three succeeded in creating a credible design for an implosion-style bomb.