Is there an idiom or phrase associated with a time when a person has little hope and has to choose between bad and worse options?
I think you are referring to the choice between the:
the less bad thing of a pair of bad things.
- I didn't like either politician, so I voted for the lesser of two evils. Given the options of going out with someone I don't like and staying home and watching a boring television program, I chose the lesser of the two evils and watched television.
(McGraw-Hill Dictionary of American Idioms and Phrasal Verbs)
"between the devil and the deep blue sea" means between two dangers and by avoiding one there's danger of falling into another.
from Wikipedia - "Between the devil and the deep blue sea" is an idiom meaning a dilemma—i.e., to choose between two undesirable situations (equivalent to "between a rock and a hard place").
from TFD - if you are between the devil and the deep blue sea, you must choose between two equally unpleasant situations.
"If I pay the rent I won't have any money for food. I'm really between the devil and the deep blue sea."
"For most people a visit to the dentist is like a choice between the devil and the deep blue sea - if you go you suffer, and if you don't go you suffer."
More poetically, there's "between Scylla and Charybdis". This is a reference to a passage in The Odyssey in which the ship had to traverse a narrow strait between a whirlpool and the lair of a monster.
See the Wikipedia article.
on the horns of a dilemma
If you are on the horns of a dilemma, you are faced with a choice between two equally unpleasant options.
- I'm on the horns of a dilemma; I have to choose between a boring job with a good salary or a more interesting job with a lower salary.
The original dilemma in rhetoric was a device by which you presented your opponent with two alternatives; it didn’t matter which one he chose to respond to — either way he lost the argument. When you did this to your opponent you were said to present two horns to him, as of a bull, on either of which he might be impaled.
Here are three more phrases to describe the fix you are in.
Between a rock and a hard place (Bloomsbury)
To be in a very difficult situation and to have to make a hard decision between two things that are equally unpleasant.
In extremis (formal) (Cambridge)
in an extremely difficult situation: I'll only ask the bank for a loan in extremis.
the end of the line (Cambridge )
the point at w#hich it is no longer possible to continue with a process or activity: We've struggled on for as long as we could, but now we're at the end of the line.
Between a rock and a hard place [commonly equated to Catch 22]
Damned if you do, damned if you don't.
(German for "compulsion to move") is a situation found in chess and other games wherein one player is put at a disadvantage because they must make a move when they would prefer to pass and not to move. The fact that the player is compelled to move means that his position will become significantly weaker. A player is said to be "in zugzwang" when any possible move will worsen his position.
Out of the frying pan, into the fire.
Sophie's Choice is a 1979 novel by American author William Styron. It concerns the relation between three people sharing a boarding house in Brooklyn: [...]
[Sophie] specifically relates her attempts to seduce Höss [the commander of Auschwitz concentration camp] in an effort to persuade him that her blond, blue-eyed, German-speaking son should be allowed to leave the camp and enter the Lebensborn program, in which he would be raised as a German child. She failed in this attempt and, ultimately, never learned of her son's fate.
Faced with a double bind
All citations from Wikipedia, list taken from Wikipedia entry for Between a rock and a hard place.
least worst [only before noun]
the least worst choice is the best choice from a list of choices that you think are all bad
Often it's a question of choosing the least worst option.
Probably not an exact response but may fit the situation:
Half a loaf is better than none 
From the Cambridge Dictionary of American Idioms, the idiom means:
Getting less than what you wanted is better than getting nothing
in which, getting less than what you wanted could be considered as bad against getting nothing or worse.
On a hiding to nothing.
To be faced with a situation which is pointless, as a successful outcome is impossible
I would suggest two approaches to the splitting of efforts, resulting in inevitable failure:
"He doesn't have a snowballs' chance in Hell..." indicating an overwhelmingly impossible task(s)
Or, owed to the Star Trek lexicon:
"He failed his 'kobayashi maru'" referring to the fictional no-win test of character from Starfleet Academy
Hobson's choice-The choice of taking what is offered or nothing at all.