How do I determine the engine thrust required to give a specific climb rate?
How do I know that the N1 that I get from the OPT is enough thrust to meet the required climb gradient in feet per minute published on the SID?


You can't, but read on please.
Variables
A few things out of the way first. A percent N1 is not a percentage of the engine's max thrust. 95% N1 at 20°C at sea-level is different from 95% N1 at 35°C at a 4,000 feet MSL airport in terms of thrust force.
Likewise for the lift; airport elevation, temperature, and pressure, also affect the lift. So for the same runway and SID, you'll get different climb rates at a given speed on different days. And since lift counters weight in a steady climb, how heavy you are plays an important role (as well as how the plane is loaded, i.e., center of gravity).
Gradient and OPT
A SID that shows a climb rate, will also show the corresponding ground speed (put together that's a gradient), in fact it will be multiple rates vs. speeds. Climb rate on its own is not a performance parameter. And while an angle of climb (flight path wrt air) does not depend on the wind, the climb gradient (wrt ground) is affected by the wind (yet another variable). Another variable, turns (bank angle) while climbing.
I'm not sure what OPT is, if it is the free Onboard Performance Tool app from Boeing on the App Store, then note that it requires contacting a Boeing representative for the configuration files, etc. If it is an OPT-like software in the sim, then I highly doubt it can calculate the climb rate; even relatively expensive performance calculators for PC flight sims don't go there.
If you have all the variables, you still can't know the gradient or climb rate at a speed. While a real Boeing 737 comes with an AFM or FPPM (manuals that include performance data), climb gradients with all engines operative are typically not included. The FAA AC 120-91 (Airport Obstacle Analysis) discusses this point:

(...) It is recognized that many AFMs generally contain only the one-engine-inoperative performance for loss of an engine at V1 on takeoff. All-engines-operating performance must also be considered to determine the airplane's flightpath in the event of an engine failure at a point on the flightpath after V1. The best available all-engines-operating data should be used consistent with best engineering practices. Operator's may find appropriate acceptable data in various sources, such as: community noise documents, performance engineer's handbooks, flight characteristics manuals, and manufacturers' computer programs.

How is it done?
So what do you do? Unfortunately not much. In the real world a program like the Boeing Climbout Program (BCOP) -- an example of the aforementioned &quot;manufacturers' computer programs&quot; -- is used to analyze each SID individually (this is different from the three or four takeoff limiting weights pilots are accustomed to).
But even then the analysis may not be presented to the pilots. The airline that is operating out of airport x -- your day-to-day typical not-in-a-hot-valley airport -- would have already checked the extremes (temperature, weight, etc.) against the SIDs for its fleet, and concluded early-on that airport x is not limiting.
The pilot will normally be presented with runway specific thrust setting / weight limit. If you're below this weight, then you may have more than one thrust/flaps setting to choose from. SIDs for the most part (for an airline flying a 737) are not limiting except in extreme airports. Note that the SIDs are designed with the planes/airlines that use them in mind, especially that noise is another important factor.
If you are not an airline pilot and you have your own business jet, then this is a different matter. But because business jets have limited payload/fuel compared to jet-liners, the performance tables are not as complex and indeed can be printed. In other words, those SID figures are more important to general aviation pilots (you may also notice that some SIDs are for turbine aircraft only).
Flight sim
For the purposes of a flight simulator, you can take this as an opportunity to test the impact of each condition one by one. If your 737 allows VNAV arming on ground, you may get &quot;unable next altitude&quot; if the plane thinks it can't make the first restriction, although this is not good practice, as the actual climb is divided into segments, each with a different performance, and the first waypoint may not be representative of the most demanding segment.
You can also test all the extremes at once and see what vertical speed you get. 1100 FPM can be achieved by a 737 in a wide range of conditions, if it can't get to 1100 FPM close to the ground, then getting to cruise will be quite the challenge, i.e., test the conditions to put that value in perspective.
I recommend browsing the aircraft-performance and performance-calculation tags, as well as checking the Airbus publication 'Getting to grips with aircraft performance'.