In a medium saucepan over medium heat, cook strawberries with sugar, stirring occasionally, until berries give up their juices and are soft, 4 to 5 minutes. Remove from heat and stir in lemon juice. With a potato masher or back of a fork, smash strawberries into a coarse purée. Transfer to a bowl and chill until cold, about an hour.
Line a 5- by 9-in. loaf pan (ideally straight sided) with a 9-in.-wide piece of foil, long enough to fit into pan crosswise and hang over edges. Remove ice cream from freezer and let soften slightly (but not to the point of soft-serve). Scoop into a large bowl, then drizzle with strawberry purée. With just a few strokes, fold purée into ice cream to maintain distinct streaks of strawberry. Spoon mixture into prepared pan and freeze until set but not rock hard, 1 1/2 hours.
Remove pan from freezer and, using foil overhang, lift ice cream onto a work surface; peel back sides of foil. Score ice cream block crosswise to mark 6 bars, then insert popsicle sticks a little over halfway into each bar. Freeze until just firm, 1 hour.
Meanwhile, crush cookies and freeze-dried strawberries coarsely with your hands, then chop evenly until crumbs are the size of large peppercorns. Strain in a fine-mesh strainer to remove dust.
Slice ice cream block through score lines to separate into popsicles. Let thaw until melted just on surface. Working with 1 bar at a time, pat cookie-strawberry crumbs all over it. Serve immediately, or set on a plate and return to freezer while you coat remaining pops.
*If ice cream is airy, you may need as much as 3 1/2 cups, because it will collapse when measured. Find freeze-dried strawberries at Whole Foods Market and online (store airtight at room temperature).
Make ahead: Crumb coating, up to 1 day, airtight at room temperature. Coated pops, frozen airtight, up to 2 weeks.