What you write is a compilation of many common misconceptions about these tests.
The short answer is: use the t test with Welch correction. Now, the details.
I would like to test whether mean (or median) answers in both groups are significantly different.
Means and medians are different things. What people usually do, is that they think in terms of means, not medians, so by default this is also what you should aim for. The Likert scale was invented by the guy Rensis Likert precisely with the intention to make it useful for computing means (not only medians). See the James Carifio, Rocco Perla, "Resolving the 50-year debate around using and misusing Likert scales" in commentaries in Medical Education, 2008, Blackwell Publishing Ltd.
The problem is that I can not use parametric (t-test) neither nonparametric (Mann-Whitney) test since data are nonnormal (thus t-test is not appropriate) (...), multimodal (...)
Definitely not! Both tests are robust with respect to the shape of the distribution. Only Mann-Whitney (it is usually called Wilcoxon-Mann-Whitney, or WMW test) requires both distributions to have the same shape.
(...) ordinal and thus not continuous (...)
"Ordinal variable" means "arithmetic means doesn't make sense on it". Like education measured in 3-level scale "1 - grammar school", "2 - college" and "3 - university". This does not imply it is not continuous (although usually it is the case).
Neither the WMW nor the t-test require continuous variables.
(...) with different variances (...)
When you use the t-test with Welch's correction (Welch B.L. The generalization of Student's problem when several different population variances are involved, Biometrika, 34, 28-38, 1938) than you don't need to worry about the unequal variances (and shapes). The t-test (as with any test based on means) is already very robust with respect to departures from normality (see e.g. Michael R. Chernick, Robert H. Friis "Introductory Biostatistics for The Health Sciences", Willey Interscience 2003 and many other books). This property comes from the fact, that it is based on means. By the virtue of the Central Limit Theorem, the distribution of the mean very quickly converges to normal distribution.
(...) and with different shapes (thus Mann-Whitney test is not appropriate).
Yes, you've got it right. Technically speaking, the Mann-Whitney U test does not test for median, but whether one distribution is offset from the other, which is something subtly different. In particular, it makes this test sensitive to differences in distribution between groups. (see Morten W. Fagerland and Leiv Sandvik "The Wilcoxon-Mann-Whitney test under scrutiny", John Willey & Sons, 2009). These differences can translate e.g. into differences in variance or skewness. So this test, in contrast to the Welch test (the t-test with Welch modification), is not safe for when there is no variance homogeneity.