This is what the final code looks like. It requires that I not only send the block list element's key as a hidden field, but also the key of the containing parent document as a second hidden field. Let me know if you can think of a simpler way to do this.

            var parentPage = umbracoHelper.Content(parentPageKey);
            foreach (var property in parentPage!.Properties)
            {
                if (property.PropertyType.DataType.EditorAlias == "Umbraco.BlockList")
                {
                    foreach (var item in property.Value<BlockListModel>(null!)!)
                    {
                        if (item.Content is FormElement && item.Settings is FormSettingsElement)
                        {
                            FormSettingsElement formSettingsElement = (FormSettingsElement)item.Settings;
                            if (formSettingsElement.Key == formSettingsKey) // If there are two forms in the blocklist, making sure this is the one that was submitted
                            {
                                string toAddress = formSettingsElement.RecipientEmailAddress!;
                                string formName = formSettingsElement.FormName!;
                            }
                        }
                    }
                }
            }