Wire Ladill domain picker and purchase modal into hosting flows.

Use the Domains API for owned domains across panel, account, and order forms, with an embedded iframe purchase modal instead of redirecting to domains.ladill.com.

Co-authored-by: Cursor <cursoragent@cursor.com>

resources/views/partials/search-screen.blade.php

@@ -14,7 +14,7 @@
         <div class="px-4 pb-3">
             <div class="relative">
                 <svg class="pointer-events-none absolute left-4 top-1/2 h-5 w-5 -translate-y-1/2 text-slate-400" fill="none" stroke="currentColor" stroke-width="2" viewBox="0 0 24 24"><path stroke-linecap="round" stroke-linejoin="round" d="m21 21-5.197-5.197m0 0A7.5 7.5 0 1 0 5.196 5.196a7.5 7.5 0 0 0 10.607 10.607Z"/></svg>
-                <input x-ref="input"
+                <input data-ladill-search-input  x-ref="input"
                        type="text"
                        x-model="query"
                        @focus="onFocus()"
@@ -34,7 +34,7 @@
             <h1 class="text-2xl font-semibold text-slate-900">{{ $heading }}</h1>
             <div class="relative mt-3">
                 <svg class="pointer-events-none absolute left-4 top-1/2 h-5 w-5 -translate-y-1/2 text-slate-400" fill="none" stroke="currentColor" stroke-width="2" viewBox="0 0 24 24"><path stroke-linecap="round" stroke-linejoin="round" d="m21 21-5.197-5.197m0 0A7.5 7.5 0 1 0 5.196 5.196a7.5 7.5 0 0 0 10.607 10.607Z"/></svg>
-                <input x-ref="input"
+                <input data-ladill-search-input  x-ref="input"
                        type="text"
                        x-model="query"
                        @focus="onFocus()"