Try clickthrough to edit template part children.

packages/block-editor/src/components/block-list/insertion-point.js

@@ -273,13 +273,15 @@ export default function useInsertionPoint( ref ) {
 		selectedClientId,
 		selectedRootClientId,
 		getBlockListSettings,
+		isPopoverHidden,
 	} = useSelect( ( select ) => {
 		const {
 			isMultiSelecting: _isMultiSelecting,
 			isBlockInsertionPointVisible,
 			getBlockInsertionPoint,
 			getBlockOrder,
 			getBlockListSettings: _getBlockListSettings,
+			isInsertionPointPopoverHidden,
 		} = select( blockEditorStore );
 
 		const insertionPoint = getBlockInsertionPoint();
@@ -291,6 +293,7 @@ export default function useInsertionPoint( ref ) {
 			isInserterVisible: isBlockInsertionPointVisible(),
 			selectedClientId: order[ insertionPoint.index - 1 ],
 			selectedRootClientId: insertionPoint.rootClientId,
+			isPopoverHidden: isInsertionPointPopoverHidden(),
 		};
 	}, [] );
 
@@ -399,7 +402,8 @@ export default function useInsertionPoint( ref ) {
 
 	return (
 		! isMultiSelecting &&
-		isVisible && (
+		isVisible &&
+		! isPopoverHidden && (
 			<InsertionPointPopover
 				clientId={
 					isInserterVisible ? selectedClientId : inserterClientId

packages/block-editor/src/store/actions.js

@@ -1391,3 +1391,15 @@ export function setHasControlledInnerBlocks(
 		clientId,
 	};
 }
+
+export function hideInsertionPointPopover() {
+	return {
+		type: 'HIDE_INSERTION_POINT_POPOVER',
+	};
+}
+
+export function showInsertionPointPopover() {
+	return {
+		type: 'SHOW_INSERTION_POINT_POPOVER',
+	};
+}

packages/block-editor/src/store/reducer.js

@@ -1727,6 +1727,16 @@ export function highlightedBlock( state, action ) {
 	return state;
 }
 
+export function insertionPointPopoverHiddenStatus( state = false, action ) {
+	if ( action.type === 'HIDE_INSERTION_POINT_POPOVER' ) {
+		return true;
+	}
+	if ( action.type === 'SHOW_INSERTION_POINT_POPOVER' ) {
+		return false;
+	}
+	return state;
+}
+
 /**
  * Reducer returning the block insertion event list state.
  *
@@ -1773,5 +1783,6 @@ export default combineReducers( {
 	hasBlockMovingClientId,
 	automaticChangeStatus,
 	highlightedBlock,
+	insertionPointPopoverHiddenStatus,
 	lastBlockInserted,
 } );

packages/block-editor/src/store/selectors.js

@@ -2194,6 +2194,10 @@ export const __experimentalGetActiveBlockIdByBlockNames = createSelector(
 	]
 );
 
+export function isInsertionPointPopoverHidden( state ) {
+	return state.insertionPointPopoverHiddenStatus;
+}
+
 /**
  * Tells if the block with the passed clientId was just inserted.
  *

packages/block-library/src/template-part/edit/index.js

@@ -42,12 +42,16 @@ export default function TemplatePartEdit( {
 	// Set the postId block attribute if it did not exist,
 	// but wait until the inner blocks have loaded to allow
 	// new edits to trigger this.
-	const { isResolved, innerBlocks, isMissing, area } = useSelect(
+	const { isResolved, innerBlocks, isMissing, area, isSelected } = useSelect(
 		( select ) => {
 			const { getEditedEntityRecord, hasFinishedResolution } = select(
 				coreStore
 			);
-			const { getBlocks } = select( blockEditorStore );
+			const {
+				getBlocks,
+				isBlockSelected,
+				hasSelectedInnerBlock,
+			} = select( blockEditorStore );
 
 			const getEntityArgs = [
 				'postType',
@@ -69,6 +73,9 @@ export default function TemplatePartEdit( {
 				isResolved: hasResolvedEntity,
 				isMissing: hasResolvedEntity && ! entityRecord,
 				area: entityRecord?.area,
+				isSelected:
+					isBlockSelected( clientId ) ||
+					hasSelectedInnerBlock( clientId, true ),
 			};
 		},
 		[ templatePartId, clientId ]
@@ -157,6 +164,7 @@ export default function TemplatePartEdit( {
 			) }
 			{ isEntityAvailable && (
 				<TemplatePartInnerBlocks
+					isSelected={ isSelected }
 					tagName={ TagName }
 					blockProps={ blockProps }
 					postId={ templatePartId }

packages/block-library/src/template-part/edit/inner-blocks.js

@@ -8,14 +8,16 @@ import {
 	__experimentalUseEditorFeature as useEditorFeature,
 	store as blockEditorStore,
 } from '@wordpress/block-editor';
-import { useSelect } from '@wordpress/data';
+import { useSelect, useDispatch } from '@wordpress/data';
+import { useState, useEffect } from '@wordpress/element';
 
 export default function TemplatePartInnerBlocks( {
 	postId: id,
 	hasInnerBlocks,
 	layout,
 	tagName: TagName,
 	blockProps,
+	isSelected,
 } ) {
 	const themeSupportsLayout = useSelect( ( select ) => {
 		const { getSettings } = select( blockEditorStore );
@@ -46,5 +48,27 @@ export default function TemplatePartInnerBlocks( {
 			alignments: themeSupportsLayout ? alignments : undefined,
 		},
 	} );
-	return <TagName { ...innerBlocksProps } />;
+
+	const [ isHovered, setIsHovered ] = useState( false );
+	const {
+		hideInsertionPointPopover,
+		showInsertionPointPopover,
+	} = useDispatch( blockEditorStore );
+
+	useEffect( () => {
+		if ( isHovered && ! isSelected ) {
+			hideInsertionPointPopover();
+		} else {
+			showInsertionPointPopover();
+		}
+	}, [ isSelected, isHovered ] );
+
+	return (
+		<div
+			onMouseEnter={ () => setIsHovered( true ) }
+			onMouseLeave={ () => setIsHovered( false ) }
+		>
+			<TagName { ...innerBlocksProps } />
+		</div>
+	);
 }

packages/block-library/src/template-part/editor.scss

@@ -76,4 +76,19 @@
 			}
 		}
 	}
+
+	&:not(.has-child-selected):not(.is-selected):not(.is-placeholder) {
+		* {
+			pointer-events: none;
+			user-select: none; // Stops text being inadvertently highlighted when selecting a template part
+		}
+	}
+
+	&.is-selected:not(.is-placeholder) {
+		background: rgba(#007cba, 0.1); // todo: use var
+	}
+
+	&.is-hovered:not(.is-selected) {
+		background: rgba($gray-900, 0.05);
+	}
 }