I've found the "issue". Isn't really one but nvm.
If someone else has the same problem with Twitter MediaSite BBcode just delete the comment arrows in the bbms_twitter_XFormer template after the installation is complete.
Should look like this:
<!--<script src="//platform.twitter.com/widgets.js" charset="utf-8"></script>
-->
<!--
If you have the twitter button deactivated and you want to use the official twitter
oembed api + twitter widget.js to build the tweet remove the comment arrows right
at the beginning and end of the first line..which would only leave the following
example: <script src="//platform.twitter.com/widgets.js" charset="utf-8"></script>
-->
<script>$(".twitter-tweet").css("border","1px solid {xen:property primaryLighterStill}");</script>
<script>$(".twitter-tweet").css("border-radius", "7px");</script>
<script>$(".twitter-tweet").css("box-shadow", "0px 0px 7px 0px {xen:property primaryDarker}");</script>
<script>$(".twitter-tweet").children("p").css("border-bottom", "3px double {xen:property primaryLighterStill}"), ;</script>