Is there a single adjective meaning 'hateful but indispensable' or something close? (Not "love-hate"). British or American would be OK.
Necessary evil comes to mind first.
a necessary evil something that you do not like but which you know must exist or happen He considers taxes a necessary evil. See also: evil, necessary Cambridge Idioms Dictionary, 2nd ed. Copyright © Cambridge University Press 2006. Reproduced with permission. retrieved online on May 22, 2014 at http://idioms.thefreedictionary.com/a+necessary+evil
I've noticed that the adjective lovable is often used to describe people or pets who elicit ambivalent feelings:
He was the bon vivant of The Boar’s Head Tavern and the philosopher of Eastcheap – a lovable rogue who not only made us laugh at him and with him, but also at ourselves [On Shakespeare's character Jack Falstaff]
GameDaily commented that while he is a "lovable oaf," others may see him as a "fat Pacific Islander who loves stuffing his face" [Description of Nintendo / cartoon character King Hippo]
I don't know how he does it but he is a lovable serial killer. [On the merits of Michael C Hall's acting performance as Dexter]
inescapable comes to mind. I think it covers your bases.
Hateful: something that you want to be removed from.
Indispensable: something that you can't do without.
Inescapable: implies that you want to escape it, but asserts that you cannot. Since we want to escape from hated things, and since we cannot escape things that are inescapable, inescapable implies that there is something that we are tied to that we wish we weren't.
The qualifying exam was an inescapable part of obtaining a PhD.
The old lady found shooting the attacking criminal inescapable: only one could survive.
I'd need to know the exact context, but I think it meets the criteria.
I'm not sure if this word carries with it the intensity you are looking for; however, nuisance tends to be used when referring to people or situations that are unavoidable. This is more of a connotative understanding of the word, but maybe this will fit your particular context?
For example, My dog is total nuisance. Your dog may agitate you, but that doesn't mean the dog is expendable.