Hey there, I don't know if someone else is faced with a similar issue.
I'm using Odoo version 15. and for disable the Send Message option on version 15 you can change the xml code on this location.

addons/mail/static/src/components/chatter_topbar/chatter_topbar.xml :

You can add simple a style in the "Send Message", I added style="display:none;"

<button class="btn btn-link o_ChatterTopbar_button o_ChatterTopbar_buttonSendMessage"type="button"t-att-class="{                                'o-active': chatter.composerView and !chatter.composerView.composer.isLog,                                'o-bordered': chatter.hasExternalBorder,                            }"t-att-disabled="chatter.isDisabled"title="Send a message"data-hotkey="m"t-on-click="chatter.onClickSendMessage">                            Send messagebutton>

In odoo13, i tried to uninstall crm_sms that did not work. So I followed Stefan's suggestion, comment the html under chatter.xml (no more mail.xml in odoo13), it works

Kind of late, but it may help some others.

For Odoo v13.​

I've achieved it by uninstalling sms module from apps, it works for all sms features in Odoo, but if you want to disable them only for CRM there's a module named crm_sms I suppose uninstalling it would disable the feature for CRM only. 

For now I've changed /usr/lib/pymodules/python2.6/openerp/addons/mail/static/src/xml/mail.xml

146c146
< <input type="checkbox" t-att-data="recipient.full_name"/>
---
> <input type="checkbox" t-att-checked="recipient.checked ? 'checked' : undefined" t-att-data="recipient.full_name"/>

This way, our users have to explicitely choose to send a message externally. It's not what you originally wanted but it might be a workaround for the interim.