An extension method that allows you to change the current search term of the searchable attribute by calling: InspectorProperty.SearchFor(SearchTerm)

InspectorPropertyExtensions.cs

#if UNITY_EDITOR
using System;
using System.Collections.Generic;
using System.Reflection;
using Sirenix.OdinInspector;
using Sirenix.OdinInspector.Editor;
using UnityEngine;

public static class InspectorPropertyExtensions
{
    private const BindingFlags PublicInstance = BindingFlags.Public | BindingFlags.Instance;
    private const BindingFlags PrivateInstance = BindingFlags.NonPublic | BindingFlags.Instance;
    private static readonly Dictionary<InspectorProperty, SearchDelegates> Delegates = new Dictionary<InspectorProperty, SearchDelegates>();

    public static void SearchFor(this InspectorProperty property, string searchTerm)
    {
        if (!property.Attributes.HasAttribute<SearchableAttribute>())
        {
            Debug.LogException(new Exception($"The property '{property.Name}' has no [Searchable] attribute applied to it."));
            return;
        }

        if (Delegates.TryGetValue(property, out var delegates))
        {
            delegates.UpdateSearch?.Invoke(searchTerm);
            delegates.ScheduleUpdate?.Invoke();
            return;
        }

        delegates = property.ChildResolver is ICollectionResolver
            ? SetupCollectionDelegates(property)
            : SetupDelegates(property);

        delegates.UpdateSearch?.Invoke(searchTerm);
        delegates.ScheduleUpdate?.Invoke();

        Delegates.Add(property, delegates);
    }

    private static SearchDelegates SetupCollectionDelegates(InspectorProperty property)
    {
        var collectionDrawer = property.GetActiveDrawerChain().Current;
        var listDrawerConfigInfo = collectionDrawer.GetType().GetField("info", PrivateInstance).GetValue(collectionDrawer);
        var filteredChildren = listDrawerConfigInfo.GetType().GetField("FilteredChildren", PublicInstance).GetValue(listDrawerConfigInfo);
        var searchFilter = filteredChildren.GetType().GetField("SearchFilter", PublicInstance).GetValue(filteredChildren);

        searchFilter.GetType().GetField("Property", PublicInstance).SetValue(searchFilter, property);

        var updateSearch = (Action<string>)Delegate.CreateDelegate(typeof(Action<string>), searchFilter, "UpdateSearch");
        var scheduleUpdate = (Action)Delegate.CreateDelegate(typeof(Action), filteredChildren, "ScheduleUpdate");

        return new SearchDelegates()
        {
            UpdateSearch = updateSearch,
            ScheduleUpdate = scheduleUpdate,
        };
    }

    private static SearchDelegates SetupDelegates(InspectorProperty property)
    {
        property.Tree.SetSearchable(true);
        var searchFilter = typeof(PropertyTree).GetField("searchFilter", PrivateInstance).GetValue(property.Tree);
        var updateSearch = (Action<string>)Delegate.CreateDelegate(typeof(Action<string>), searchFilter, "UpdateSearch");

        return new SearchDelegates()
        {
            UpdateSearch = updateSearch,
            ScheduleUpdate = null,
        };
    }

    private struct SearchDelegates
    {
        public Action<string> UpdateSearch;
        public Action ScheduleUpdate;
    }
}
#endif