Thank you for the help.
The issue apparently was not in web parts but rather IE10+ compatibility issues with image buttons for which after adding a script the buttons started working.
<script type="text/javascript">
Sys.WebForms.PageRequestManager.getInstance()._origOnFormActiveElement = Sys.WebForms.PageRequestManager.getInstance()._onFormElementActive;
Sys.WebForms.PageRequestManager.getInstance()._onFormElementActive = function (element, offsetX, offsetY) {
if (element.tagName.toUpperCase() === 'INPUT' && element.type === 'image') {
offsetX = Math.floor(offsetX);
offsetY = Math.floor(offsetY);
}
this._origOnFormActiveElement(element, offsetX, offsetY);
};
</script>
