endConversation() method from lightning-conversation-toolkit-api in the Salesforce Lightning Web Component can be used to create custom End Conversation Button for Salesforce Enhanced Channels like Messaging for In-App and Web.

Sample Lightning Web Component:

HTML:

<template>
    <lightning-conversation-toolkit-api lwc:ref="lwcToolKitApi">
    </lightning-conversation-toolkit-api>
    <lightning-card>
        <lightning-button 
            label="End Conversation" 
            onclick={endConversation} >
        </lightning-button>
    </lightning-card>
</template>

JavaScript:

import { LightningElement, api } from 'lwc';

export default class EndConversation extends LightningElement {

    @api recordId;

    async endConversation( event ) {
        
        const toolKit = this.refs.lwcToolKitApi;
        let result = await toolKit.endConversation(
            this.recordId
        );
        console.log(
            'result is',
            result
        );

    }

}

js-meta.xml:

<?xml version="1.0" encoding="UTF-8"?>
<LightningComponentBundle xmlns="http://soap.sforce.com/2006/04/metadata">
    <apiVersion>60.0</apiVersion>
    <isExposed>true</isExposed>
    <targets>
      <target>lightning__RecordPage</target>
    </targets>
</LightningComponentBundle>