As you reported, Wild Shape says
[...] your ability to speak or take any action that requires hands is limited to the capabilities of your beast form
to say something very quietly so that other people cannot hear you
A bear can neither speak nor whisper, hence your DM was right in this ruling. Moreover, the Sage Advice Compendium supports your DM's decision:
Can a Circle of the Moon druid speak the languages it knows while in the form of an elemental? Yes, since the elementals listed in Elemental Wild Shape can speak. A literal interpretation (RAW) of Wild Shape could reasonably lead you to think that transformed druids can speak only languages that appear in an elemental’s stat block, but the intent (RAI) is that druids retain their knowledge, including of languages, when they transform and can speak the languages they know if an adopted form can speak.
You can come up with a strategy to overcome this problem: you may decide with the other members of your group a sort of simple code for communicating via Message while you are wildshaped. For example, in the case of your being a bear, you can decide that a low growl means "yes", two quick low growls mean "no" and so on. This is up to you and mainly to your DM.
Another way is to wildshape into a raven (available at 8th level):
Mimicry. The raven can mimic simple sounds it has heard, such as a person whispering, a baby crying, or an animal chittering. A creature that hears the sounds can tell they are imitations with a successful DC 10 Wisdom (Insight) check.
Moreover, there are several animals\$^1\$ that speaks their very unique language: Giant Owl, Gian Elk and Giant Eagle. Problem is that the caster of Message should be able to understand those languages.
\$^1\$ Thanks to @Kogarashi Kaito for these further examples.