You may be able to get over the initial current bump with a big capacitor but you're still going to have a significant current draw from the solar.
The other thing is, you would be a lot better off supplying the ESP unit with 3.3V - I think this was mentioned before. The Wemos D1 takes up to 5V but then immediately steps it down to 3.3V using a very inefficient LDO so you're wasting a TON of power there. If I remember correctly, you can use a 3.3V buck/boost onto the 3V3 pin and it will power the unit and completely bypass the LDO.
You could even try putting 3.3V onto the 5V pin - the LDO isn't very efficient but the ESP chip will run as low as 2.8V so you may find it works.