Using the fitting tool, I did an experiment to test out how meaningful these differences are. The constant will be the items and modules, the variables are the logistics dropsuits. This is to test the potential of the suit, I'm not saying this is how a smart player would choose to use them.
Experiment: Max skills + Duvolle AR + maximum tanking

ak.0 Logistics
3 Complex shield extenders
3 Complex armor Plates
403 shields & 604 armor (972 EHP)
Speed: movement 3.53 m/s sprint 5.18 m/s
171 CPU & 6 PG remaining

ck.0 Logistics
4 Complex shield extenders + 1 enhanced shield extender
3 Complex armor plates + 1 enhanced armor plate
618 shields & 659 armor (1277 EHP)
Speed: movement 3.74 m/s sprint 5.49 m/s
26 CPU & 0 PG remaining

mk.0 Logistics
4 Complex shield extenders
4 Complex armor plates
368 shields & 693 armor (1094 EHP)
Speed: movement 3.76 m/s sprint 5.53 m/s
100 CPU & 0 PG remaining

gk.0 Logistics
2 Complex shield Extenders + 1 enhanced shield extender
5 Complex armor plates
330 shields & 827 armor (1157 EHP)
Speed: movement 3.5 m/s sprint 5.12 m/s
142 CPU & 4 PG remaining

Ordering from best to worse
Most HP potential: ck.0, gk.0, mk.0, ak.0
Most versatile (most module slots): ck.0, mk.0, gk.0, ak.0
Fastest while tanked: mk.0, ck.0, ak.0, gk.0
Most capable of support: mk.0 & gk.0 tied for first place. gk.0 & ak.0 tied for second place.
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Conclusion:

The prototype Amarr logi sacrificed the gain of a module and the gain of an extra equipment slot to get a sidearm. I propose giving the prototype and advanced Amarr logistics 1 more module slot and appropriate PG/CPU to use it, or 1 more equipment slot to use. A sidearm is worth a module slot, OR an equipment slot, but not both.

The Caldari sacrificed the gain of 1 equipment slot for getting 2 more module slots instead of one than its assault counterpart. This combined with the 25% (when maxed out) shield extender efficacy boost per level, makes it a bit overpowered. I suggest removing the extra module slot gain, and replacing it with an equipment slot or something. Seriously, 9 module slots is just too much.

The Minmatar and Gallente logis are balanced. They are tanky support who are slower than their assault counterparts, and carry lots of equipment to do the job. No problem there.

IMPORTANT EDIT: To illustrate how terrible of a balance issue this is, consider the fact that with all skills maxed out, a SENTINEL ak.0 can only fit 1258 EHP. The Caldari logi fits 1277, that is 19 more EHP than a freakin heavy sentinel with the most HP it can fit! and guess what else, the Caldari logi with its superior EHP is FASTER than the ak.0 heavy when fitted this way; Fully tanked sentinel moves at 3.1 m/s, and sprints at 4.34 m/s.
Boundless HMG + 1 complex shield extender + 2 complex armor plates + 1 enhanced armor plate + 1 basic plate
That is 1258 total EHP, 3.31 movement speed, and 4.34 sprint speed.