sendTextMessage() and setAgentInput() methods from Salesforce lightning-conversation-toolkit-api can be used in the Lightning Web Component to send message and set message in the Enhanced Conversation Component in the Lightning Record page.

Sample Lightning Web Component:

HTML:

<template>
    <lightning-conversation-toolkit-api lwc:ref="lwcToolKitApi">
    </lightning-conversation-toolkit-api>
    <div style="width: 200px;">
        <lightning-input 
            label="Message" 
            value={message} 
            class="slds-p-around_small"
            onchange={handleMessageChange}>
        </lightning-input>
    </div>
    <br/>
    <lightning-card>
        <lightning-button 
            label="Set Message"
            onclick={setMessage}
            class="slds-p-around_small">
        </lightning-button>
        <lightning-button 
            label="Send Message"
            onclick={sendMessage}
            class="slds-p-around_small">
        </lightning-button>
    </lightning-card>
</template>

JavaScript:

import { LightningElement, api } from 'lwc';

export default class CustomMessageComponent extends LightningElement {

    message = '';
    @api recordId;

    handleMessageChange( event ) {
        this.message = event.target.value;
    }

    async setMessage() {
        
        const toolKit = this.refs.lwcToolKitApi;
        let result = await toolKit.setAgentInput(
            this.recordId,
            { text: this.message }
        );
        console.log(
            'result is',
            JSON.stringify( result )
        );

    }

    async sendMessage() {
        
        const toolKit = this.refs.lwcToolKitApi;
        let result = await toolKit.sendTextMessage(
            this.recordId,
            { text: this.message }
        );
        console.log(
            'result is',
            JSON.stringify( result )
        );

    }

}

js-meta.xml:

<?xml version="1.0" encoding="UTF-8"?>
<LightningComponentBundle xmlns="http://soap.sforce.com/2006/04/metadata">
    <apiVersion>63.0</apiVersion>
    <isExposed>true</isExposed>
    <targets>
      <target>lightning__RecordPage</target>
    </targets>
</LightningComponentBundle>

Output:

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.

Custom End Conversation Button for Salesforce Enhanced Channels