While both PCR and partial least squares seem reasonable here, I'd say those are better suited to situations where you have more variables. There are other ways to deal with collinearity, including ridge regression, which removes the requirement that the estimates be unbiased in order to greatly reduce the variance of the estimates (that is one of the main negative consequences of collinearity).

However, you should also note that collinearity is not a problem if your only goal is prediction, which you say is your goal. However, you also say (correctly) that

doing a standard multiple linear regression might give bad results (for example with some negative βi where it shouldn't).

but this is a problem of explanation. That is, the parameter estimates are hard to interpret, substantively, because controlling for the other variables makes it a bit confusing.

As to the loss from using just one of the variables, you can run both regressions and see what happens. You can compare $R^2$ or adjusted $R^2$, or residuals, or whatever you like. My guess would be that excluding any of these might result in more outliers and worse predictions, but that's just a guess.