I pose just a few questions.

Are there any Imperial Laws mentioned in fluff or fiction that prohibit men from pursuing scientific understanding outside of the Adeptus Mechanicus? Are there examples supporting ether side?

In turn, are there Imperial citizen, clans, families, and organizations that are legally allowed under Imperial law to practice the manufacturing and understanding of advanced technology that are not bound by the cult of the machine or forced to reveal their full knowledge to the Priests of Mars?

Are technological Heretics truly Heretics in the eyes of the Imperium even if they are not under the corruption of chaos or wish ill of the rule of Terra, or do they fall under the Mechanicus's obligations of enforcement that are not fully supported by the Imperial Creed or the Lex Imperialis?

The reason for these questions is for insight into a Rogue Trader's plight against the Mechanicus. His ship contains more then several pieces of rare technology that he is under no obligation to release to the Mechanicus due to his status without a direct order from the lords of Terra. Thus has been the target of a Machine Cult incited plot to destroy him and his entire family line to force the acquisition of his ship. Unable to trust the Cult even though the attempt was made by a faction within the cult rather then a whole, he seeks outside aid without turning to heresy.

It is the sole and sacred purview of the Adeptus Mechanicus to handle technology. Innovation is generally a bad thing, with nearly all designs you'll see in the 41st Millenium dating back to ancient Standard template Constructs. New creations take years of fighting through paperwork to become accepted; chances are, if you come up with something clever and are anything less than a Magos, expect some sort of punishment (often being branded as Hereteks and sentenced to death). That said, some radicals are exiled without being punished and continue their independent studies as Maleteks. Maleteks are somewhat shameful, but are often used by the Mechanicus when they want something done without a paper-trail leading back to them.

However, Warhammer 40,000 is a universe of contradictions. The Enforcer series has a multitude of noble houses given ancient rights to innovate on produce on certain designs (specific weapons, in this case) as a major plot point. The Inquisition often has research done into technology (the Ordo Xenos frequently works with the Mechanicus' Order Biologis, while some radical factions pursue heretical warp-tech) and some battlefield modifications later become accepted (this is usually only accepted when the Astartes does this sort of thing).

It sounds like your Rogue Trader is in for quite a fight. My advice for him would be to pursue aid from another part of the Mechanicus (perhaps making an agreement to let them study his ship without restricting his travels?) or seek help from the Adeptus Arbites. If neither of those sound good, he should prepare to lose everything: the Adeptus Mechanicus once stole a relic from an Adeptus Astartes chapter, who in turn lead an assault to take it back. They were branded traitors to the Imperium of Man and declared Excommunicate Traitoris. If an entire chapter could suffer such a fate, then one Rogue Trader will not fare much better.

Hope this helps!

There are scientific research establishments and universities of the sciences (both hard and soft) that are outside Mechanicus control, ranging from noblemen's salons to grand planetary universities. Some of them are affiliated/allied with the mechanicus, some aren't; some of them accept mechanicus members, some don't (although the ban may well be unofficial- you don't want to do something to offend them that they can't ignore).

Examples show up in Eisenhorn (specifically in Malleus- Aemos poses as a senior research fellow of a Universitae), Innocence Proves Nothing (a local salon of philosophers, scientists etc. acts as the link for a heretical cult), and some scattered references elsewhere that I can't place offhand.

As for production and applied research… it's an iffy legal position. A strict reading of the Compact signed back at the start of the Great Crusade might grant the Martian Technocracy a complete, and perpetual monopoly on all technology throughout the Imperium and Martian territories, but such a position is unenforcable, simply because of the sheer scale of the Imperium (not to mention the question of where you draw the line between "this is science tech, it's the tech-priests' job" and "this tech is basic enough for all"- quantum computers? PCBs? intricate clockwork analytical engines? pneumatic switching machines? Steam engines? Wheels and levers?).
The logical compromise (and one that appears to have been made, on a lot of occasions) is to license certain groups to build and improve upon particular items and/or areas, and to turn a blind eye to some other groups that do the same without sanction (and then claim it was STC all along…). Of course, that hasn't stopped the Mechanicum from coming down like a tonne of rectangular building things on people or groups they believe have gone too far.
Obviously, that doesn't always work, particularly when the Astartes are involved- just look at the (ongoing) hissy fit the AdMech threw/are throwing over the Astartes RSVs (Hunter class destroyer, Nova and Gladius class frigates).

TK-4117 said:

The Fabricator General is one of the High Lords. If you go so far as to create an incident that provokes war with a substantial grouping of the Adeptus Mechanicus, you might be subject to having your Warrant revoked, That's an extreme result.

On a more local level, the Adeptus Mechanicus can simply make it hell for you to get needed supplies and (especially) labor to keep your vessels properly maintained. This can cause breakdown in the hardware and even total breakdown of morale in a very short time (So, the Gellar Field hasn't had a tune-up and is now 90% likely to hold, eh? Um, I think I'll get off here.).