You can give the following code a try. Probably a more elegant option is available, but this seems to work for the requested value cycling:
<$button>
<$action-setfield $tiddler=<<currentTiddler>> class={{{ [<currentTiddler>!has[class]then[multicol2]] ~[<currentTiddler>class[multicol2]then[multicol3]] ~[<currentTiddler>class[multicol3]then[multicol]] }}} />
<$list filter="[<currentTiddler>class[multicol]]" variable="deleteClass">
<$action-deletefield class />
</$list>
<$list filter="[<tv-config-toolbar-icons>match[yes]]">
{{$:/core/images/spiral}}
</$list>
<$list filter="[<tv-config-toolbar-text>match[yes]]">
<span class="tc-btn-text">
Display
</span>
</$list>
</$button>