Does useful load increase with more horsepower?
First, you ask one thing in the title and another in the text of your question. Useful load is the sum of fuel and payload, whereas the gross weight is the force produced by the mass of the whole aircraft in the gravitational field of Earth.

First: Gross Weight

If you change engine power during airplane design, gross weight will go up. At that point, you will have relative parameters like wing loading (kg/m²) and power loading (kW/kg) defined and will size the aircraft accordingly.

If you switch the engine of an existing aircraft to a more powerful one, you will need to modify the airframe and systems if the aircraft had not been designed for the more powerful engine initially. Now your gross weight is already defined and cannot simply be changed. Let's first see what happens if a more powerful and heavier engine is fitted:


The higher engine mass will affect the location of the center of gravity and reduce the possible payload. Note that the mass increase will include a stronger engine mount and firewall, and possibly a balancing weight in the tail to put the center of gravity back into the legal range.
If you want to maintain the same power loading, you will need to increase gross weight, which in turn will increase structural loads. Now you need a heavier structure.


For now, let's assume that you stick with the old gross weight and reduce payload accordingly, because you do not want to rivet doublers all over the structure. Fine, but we are not done yet.


You might want to fit a larger propeller, which might require changes to the landing gear to keep a minimum ground clearance.
Your minimum flight speed will go up. If the aircraft is certified according to FAR part 23, the minimum speed should stay below 61 knots (see §23.49(d)). Maybe you will need to fit more effective flaps to keep the minimum speed constant.
Your maximum flight speed will go up. Technically, you can keep the limit speeds of the less powerful aircraft, but dive speed is calculated in one case by assuming 75% engine power (see §23.335(b)4i), so you will most likely be required to adjust limit speeds.
Once you raise limit speeds, structural loads go up. Now you need again to start riveting doubles all over the place.


You see, you cannot legally escape. Fitting a more powerful engine will require adaptions in the structure, and once you need to reinforce wings, landing gear and fuselage, you can as well choose a higher gross weight. But now you need to re-size the structure and re-certify the modified aircraft.

And we have not even started to look at the systems:


The bigger engine will have a higher fuel flow. Are the fuel pumps adequate, or will you need to switch them for more powerful pumps?
The higher design speeds might increase control forces, so now you need to modify the control system to avoid exceeding maximum stick forces (§23.143, §23.175)
Will the new engine excite new harmonics somewhere in the aircraft? Maybe now a part will vibrate and shake itself to pieces which had stayed calm and in one before?


The list goes on, but this shall suffice.

Now to your specific case: If you want to keep the power loading constant (which will give you the same performance as before), the airplane gross mass goes up by 54%, and so does the wing area.

If you want more performance, you will need to reinforce the structure (or start all over with a new design of the same size), and now it is up to you which gross weight to pick. The heavier engine and structure will eat into your payload and fuel, but if you accept that limitation, you can fly with the same gross weight as before. 

Second: Useful load

As you can see from the first part of the answer, useful load is reduced by fitting a heavier and more powerful engine. Only when you re-work the structure to cope with the higher loads can you raise wing loading and regain the loss in useful load. Whether you can increase it by more than what you have lost so far depends on the maximum wing loading.

Let's assume for the rest of the answer that the outside shape (wing area, airfoil etc.) will stay identical to the original aircraft, except for maybe a bigger cowling, propeller and radiator.

The mass increase of the power plant plus its supporting structure is linear to the increase in performance. To beef up the structure, you could use the same percentage rule to stay on the safe side, but the real increase will be smaller. There are books which list parametric formulas for airplane masses, and those will show you the influence of the main parameters. They show that wing mass goes up with the square root of dive speed. Since speed is proportional to the third root of engine power, you will see that 54% more power means 15.5% more speed and 7.5% more structural mass.

Now let's assume that the propulsion system is 15% of the aircraft mass, and structural mass is another 40%. Your empty mass will go up by 8% (engine) and 3% (structure), and if your useful load was 30% of gross weight before, it will now be down to 19% of the same mass. Now we need to bump up the gross mass, and go for half of what you would gain if you keep power loading constant. This means your gross mass is going up to 127% of the original gross mass.

Before you rejoice that now your useful load has gone up to 46%, we need to account for the additional structural reinforcement that this gross mass increase produces. The parametric formulas give a structural mass increase proportional to m$_{gross}^{0.3}$ for the fuselage and m$_{gross}^{0.7}$ for the wing, and a higher wing loading will again result in a dive speed increase, so your total structure will now maybe weigh 15% more than before, for a total of 47% or 48% of the initial gross mass. Yes, your structural mass fraction will go up when you  increase wing loading! But still a sizable increase in useful mass should remain - you will now have 39% or 40% of the original gross mass available for fuel and payload, for a relative increase of 30% over the original useful load.

Now I never tried to keep minimum speed constant, but that would had complicated this answer even more. If the original plane had no or very simple flaps, this would be quite feasible, but result in almost a new wing design.