One thing to consider is that Inquisitor has a literal "Emperors Permission to Do Whatever Needed" and in terms of theoretical "ranking" in Imperium Inquisitorial Mandate outranks anyone and everyone save the Big E himself.
If required, Inquisitors may call on the service and/or resources of any Imperial servant or organization. Not even a High Lord of Terra may refuse the order of an Inquisitor without good reason. This power extends across the Adeptus Astartes and the Adeptus Mechanicus, however learned Inquisitors show discretion and request the assistance of the Space Marines and attempt not to anger the Adepts of Mars.
The Space Marine chapters autonomy depends on their mandate to protect the mankind by any means they deem necessary. There has been several historical examples of Space Marine chapters clashing with other Imperial organizations (including Inquisition) over if they need to answer to anyone else. Save for a few rare examples all Space Marine chapters who have openly challenged Inquisitorial Mandate have ended up Excommunicate Traitoris and destroyed by other marines.
In Deathwatch there really should not be any issue over this: The relationship of Chapters, Deathwatch and Inqusition is quite clear. Deathwatch IS part of Inquisition and as such once you have sweared the oath and joined the Deathwatch your original chapter has been outranked and no-one from your original chapter has any say over you untill you end your Deathwatch tour. How your chain of command goes inside Deathwatch varies, but it should go without questioning that Watch Captains, Watch Commanders and all Inquisitors outrank you by a good few miles.
If you have a problem with this particular Inquisitor then the best and most prudent course of action is to nod, smile and say yes... and keep eye on her. After all, you do have your own Watch Captain, who is your direct superior, to report to after everything is over. If the mission was kosher and went well, then what did you lose? Nothing. If it went south, report to Watch Captain as he certainly has much more influence with Deathwatch and Inquisitors than you can ever dream of.
Whatever you do, DO NOT even think about murdering the Inquisitor just because you don't like her manners. She technically has the direct mandate from Big E and even if you do manage to kill her off you have just broken your oath to Deathwatch and the Watch Captain WILL punish you if they find out. He has no other choice, really, since if they let any Space Marine go around killing Inquisitors on whim they risk having the whole Deathwatch/Ordo Xenos deal fall to ruins, resulting in xenos getting upper hand and Deathwatch as an organization failing in their sworn duty towards Emperor and Mankind.
In short: If you start to kill your own people (Inquisitors most certainly included) on whim you might as well start looking for navigation maps to Eye of Terror and a nice Chaos God to swear your allegiance to, because that is EXACTLY where that path will lead to...