Let's get a few things right
Rizzo - Team control through 2021; doesn't cost more than 14.5 million
Castro - Team control through 2020; doesn't cost more than 12 million until 2020 when Cubs have one million/16 million option
Lester -Drops to 20M in 2020 and 2021 is 10M/25M team option
When does Arbitration 1 kick-in
LaStella - Arbitration 1 in 2017
Baez - Arbitration 1 in 2018
Hendricks - Arbitration 1 in 2018
Bryant - Arbitration 1 in 2018
Russell - Arbitration 1 in 2018
Schwarber Arbitration 1 in 2019
The Cubs won't have to pay for hitters until 2019 (that's year four of any deal signed this off-season) and even then, they don't have FA to deal with on these guys until 2021. And the Cubs this year are paying Edwin Jackson 13 million, Montero (might be gone) 12 million, Wada 4.5 million, so there is PLENTY of money to still have a not Dodgers-esque payroll and sign guys both now and later.
Just to show you the flexibility, they can do the following
Pay Fowler the same as what they're paying him+Wada+Motte (16 million)
Pay Arreita what they're paying him+Jackson+Monetro (30 million)
The angels signed Trout to an extension before he even hit arbitration. It's plausible to think the cubs could with at least Schwarber and Bryant on a similar timeline. They probably only are talking about $3-5 mil through your early arbitration years but that adds up. It's pretty realistic to think that Arrieta could get $30 mil a season given that Szchezer got that. Castro makes $11 mil in 2018 and basically $12 mil in 19. Lester makes $27 mil in 18 and $20 in 19. Rizzo makes $7.286 mil in 18 and $11.286 mil in 19. It's not crazy to think Fowler could get $20 mil/season. So between Fowler, Arrieta, Castro, Rizzo and Lester you're talking about $95.3 mil in 18 and $93.3 mil in 19 assuming no other commitments.
The CSN portion of the cubs TV rights runs through the end of the 2019 season IIRC. So, depending on how well they do with those you may end up with some big time cash later on but that wont be immediate. That will take a few years to come together. My guess is the cubs likely will be in the $150 mil payroll range next year and given where they are headed success wise I wouldn't be totally surprised to see them in the $160-170 mil range by 2019. But even ignoring those rookie extensions that only leaves you roughly $70 mil for the rest of the team. If you're dropping $20 mil on another starter that basically dries up any other money to do anything else because you'd need another 19 players(plus another 15 on the 40 man) for $50 mil which is doable but you're talking roughly $2.6 per player. The cubs this year after their top 6 players in salary were $2.9 per
So, is it possible they can add a $20 mil pitcher? Sure. But I don't really see it happening because teams rarely go that close to their margins. Keep in mind if the cubs wanted Shields they could have had him this past offseason for under $20 mil and in that 4 year window but they instead gave Hammel the money. Maybe you argue that this year changes expectations but I'm not entirely sure it does with regard to their money. Obviously opinions can vary but I don't really think they will sign someone longer than 4 years. Their whole mentality has been to build via the draft since Theo came on board.... well that and trades. Other than Lester and Jackson their pitching signings have been guys they hoped to fix and Bosio has done that. Hammel has been a very effective pitcher for the cubs just like Malholm and Feldman before him. It wouldn't surprise me to see them set their sights a little higher in the $10-15 mil/season and 3-4 year deal instead of the $5-10 mil/season 1-2 year deal though.
Also, keep in mind if the MLB doesn't institute a international draft by then you will have seasons where the cubs drop large portions of money into the IFA pool like this year. They have spent $3 mil on E.J. Martinez, $1.3 mil on Yonathan Perlaza $2.5 mil on Yonathan Sierra Estiwal, $2 mil on Aramis Ademan, $1.25 mil on Miguel Amaya, $600k on Yunior Perez. The cubs have a pool of $3.2 mil roughly though they traded some of that away. Anything over that is taxed double. So, the cubs are already around $7.5 mil over which will be doubled. And there's still several cuban prospects out there the cubs might still go after. Doing this takes the cubs out of the big money guys the next two years but last year under the same restrictions the cubs signed Yeiler Peguero to $250k, Wander Cabrera to $250k, Tony Rijo to $100k and Emilio Ferrebus to $250k. So even when they could only drop $250k per player they still managed to spend around $1 mil. Obviously those numbers don't seem big but when you think about the fact the cubs have basically spent $17-ish mil on players who contribute 0 to the 40 man roster right away this season(though i'm imagining some of that is out of next seasons budget) it all adds up.