Hi,

Try with the following code.

<templates>
   <t t-name="important.Chatter" t-inherit="mail.Chatter"
      t-inherit-mode="extension">
       <xpath expr="//*[contains(@class, 'o-mail-Chatter-logNote')]"
              position="replace">
           <button t-if="props.hasMessageList" class="o-mail-Chatter-logNote btn text-nowrap me-1" t-att-class="{
                    'btn-primary active': state.composerType === 'note',
                    'btn-secondary': state.composerType !== 'note',
                    'my-2': !props.compactHeight
                }" data-hotkey="shift+m" t-on-click="() => this.toggleComposer('note')">
                    Your Button Name
                </button>
      </xpath>
  </t>
</templates>

Reference:

-https://www.cybrosys.com/blog/how-to-add-a-custom-button-to-the-chatter-in-odoo-18

Hope it helps