Why do my 3 custom voltage pulses stop working as soon as I plug in the A1362 Hall-Effect sensor?
I think you are under the misapprehension that R3 and R4 define the voltage VpM when Sw9 is closed. That is not the case : R3 and R4 only define VpM as VpM = V1 * R4/(R3+R4) when Sw9 is open.

Close Sw9 and the voltage VpM is defined by R3 and the parallel combination of R4 and the output pin Vout. We don't know very much about that; the datasheet does say (on page 2) that it can sink a maximum of 10mA. Therefore you may have to assume a current source in parallel with R4 capable of sinking anything up to 10ma (i.e. sourcing up to -10ma) and designing your programming voltage sources with that in mind.

The programming instructions do refer to programmable power supplies as programming voltage sources, which suggests that low impedance sources are required.

There is also the remark about 300ma and 0.1uf being required to actually blow a fuse : the current required during the remainder of the programming operations is presumably much less than this; probably of the order of the 10ma output sink current specified.