Combine all-purpose flour, whole wheat flour, brown sugar, baking powder, and salt in a food processor and pulse to mix.
Add cold cubed butter and pulse until mixture resembles coarse crumbs with pea-sized pieces.
Add vanilla extract and about three quarters of the whisked eggs, then pulse until dough just holds together without being sticky.
Add remaining egg gradually if dough is too dry to form.
Shape dough into a rectangle, wrap in plastic, and chill in refrigerator for 30 minutes to 2 hours.
Preheat oven to 350°F (175°C).
Place strawberries, dates, spiced rum, honey, lemon zest and juice, cinnamon, cloves, nutmeg, and salt in a small saucepan and stir.
Bring filling mixture to a simmer over medium heat, reduce to low, and cook uncovered for about 15 minutes, stirring occasionally, until thickened with no bubbling liquid.
Let filling cool slightly, then blend in a food processor until mostly smooth.
Divide chilled dough into four equal parts. On a lightly floured surface, roll each into a 9-inch by 4-inch rectangle.
Pipe two parallel stripes of filling side by side down the center length of each dough rectangle, leaving short margins at the ends.
Fold dough edges over filling, pinch seams and ends tightly to seal, then place seam side down and gently roll to flatten bars evenly.
Transfer logs seam side down to a parchment-lined baking sheet.
Bake for 18 to 22 minutes until bottoms are lightly browned.
Cool bars completely on a wire rack before slicing each log into four bars.