A tough choice between two alternatives

On the horns of a dilemma

This emphasizes the difficulty of a choice. To my mind it suggests that the decision will have negative consequences in one, but not both directions. This point of view may seem at first sight difficult to reconcile with the common dictionary definition:

"to choose between two things, both of which are unpleasant or difficult"

However, applied to the example from the poster the choice between the two unpleasant things is a choice between “not going to the World Championships” and “losing a friendship”. In each case the choice results in a partial positive outcome — “going to the World Championships” or “maintaining a friendship”.

More negative in both directions (to my mind) might be the rather overused:

Between a rock and a hard place

Neither seem to me quite as negative as the “lose-lose” of @RichardKayser, although I think his “no-win” is possible.

Metaphoric use of ‘balance’ in English

There are several metaphors using balances or scales in English, but those that come to mind relate solely to the outcome (the heavier or favoured balance pan) when the equilibrium is disturbed, rather than nature of the possible outcomes (the contents of the balance pans). Thus, we have:

Finely balanced
In the balance

…relating to the uncertainty of the outcome, and:

Tip the balance in favour of…
On balance

…relating to the actual outcome. Although, somewhat counter-intuitively, the balance itself (as in financial usage) has come to mean the difference between the weights in the two pans.