One part of the question says:
Before the fly-over, Mars's rotation can be accelerated to up to 1 Martian day per hour if that helps Mars to disintegrate.
Would that help Mars break up into pieces?
Habitable Planets for Man, Stephen Ho. Dole, 1964, pp. 58-60, says that the lower limit for the length of day of a habitable planet would be when the planet rotates so fast that material on the equator reached orbital velocity.
Dole also discusses how rotation rates can affect the shapes of planets on pages 41-48.
Mars has a mean radius of 3,389.5 kilometers and an equatorial radius of about 3,396.2 plus or minus 0.1 kilometers. Thus it has an equatorial circumference of about 21,338.935 kilometers. If Mars rotated once in one Earth hour of 3,600 seconds it would rotate at about 21,338.935 kilometers per hour or about 5.9274819 kilometers per second. That is larger than the orbital velocity at the Martian surface. In fact it is also larger than the Martian escape velocity (at the surface) of 5.027 kilometers per second.
So if Mars stars rotating that fast a lot of Martian material will fly away at more than escape velocity, and a lot of other material will start to orbit the planet.
But that would not be enough to make Mars entirely disintegrate. You want no chunks of Mars larger than 1 percent of the Martian mass to remain.
If all of Mars had the same density as its overall density a body with 0.01 the volume of Mars would have 0.01 the mass of Mars. According to my rough calculations the cube root of 0.01 is approximately 0.2154444. So a spherical object with the overall density of Mars and 0.01 Mars' mass would have 0.01 of Mar's volume and thus a radius of 0.2154444 of Mars' radius and thus a radius of 730.24843 kilometers. Such an object would probably be pulled into a spheroidal shape by its gravity, though it might take a long time for a jagged Martian fragment to do so.
Since the Martian crust, mantle, and core have different average densities, crust and mantle fragments with 1 percent of the mass of Mars would be larger, and core g fragments with 0.01 percent the mass of Mass would be smaller, than calculated above.
The Martian iron nickel sulfur core is quite large, believed to have a radius of 1,794 kilometers (1,115 miles) plus or minus 65 kilometers. The Martian core is partially fluid.
The Martian core radius is more than half the radius of Mars and about half the size of the Earth's core. This is somewhat larger than models predicted, suggesting that the core contains some amount of lighter elements like oxygen and hydrogen in addition to the iron–nickel alloy and about 15% of sulfur.
Guessing that a chunk of Martian core that had 0.01 of the total mass of Mars might have a radius of only maybe 500 kilometers, twenty or thirty of them might fit within the core of Mars. Thus the Martian core might have to broken up into many smaller pieces so that none of the pieces had more than 0.01 of Mars's total mass.
There isn't much problem with breaking up the parts of the Martian core which are still fluid. If Mars spins rapidly enough, or passes close enough to jupiter, the fluid parts of the Martian core will find themselves in orbit around the remnant of Mars or even escaping from Mars.
But the solid inner core of Mars would be a problem. It would have originally been molten and gradually cooled to solid state over billions of years. So it would be essentially a single giant piece of solid metal and other materials. It would not be held together merely by its gravity but also by electrochemical attraction between its atoms and molecules. And those bounds might be harder to break than those of gravity and require greater stress.
If Mars does break up and the pieces go into orbit around Jupiter, it would become like a super asteroid belt for space mining.