jaimehlers: If you have the chest selection menu open, you can 'skip' the chest drop animation. In addition, under settings you can disable 'toggle pausing on popups' so you can open chest without the game stopping.... And another hint: You might want to turn low quality mode on in the menu, it also makes that challenge a bit easier.
My three tips to beat this: 1# Start the game mostly covering only 1 quarter of the web (from middle to the edge, then interconnect them with lines) and make it as dense as possible. Keep doing this until you have entire map covered 2# When you see a firefly or bee, hit space and eat it ASAP. 3# Keep your spider actions queued up (and do it intelligently, for example do not eat two blue bugs in a row)