hm.. solution is not as easy as I first thought.
You mentioned needing an off-the-shelf certified device. Is that for a specific reason? I can think of multiple ways to handle your problem, some of which have already been mentioned, but I can't think of any off-the-shelf solutions that are calibrated and have your specific requirement (let's call it DC crest factor) in the specs.
There are two ways to deal with signal like yours--analog integration and the digitizing approach. With digitizing, the sample rate affects your accuracy as much as the sample accuracy. So to measure the 1ms spike, you need to digitize at 100kSa/s to get 1% accuracy. I think any digitizing system set up to catch your spikes is going to be totally inaccurate as to the residual 4uA current.
I realize you don't need to actually measure the average current to 5ppm. What you need is a reasonably accurate analog integration system that will simply add the charge from the spikes into a longer period without such a large swing in the signal voltage presented to the ADC of whatever instrument you are using. As mentioned, DMMs have that, but since there are typically gain and buffer stages before the integrator part, there is a limit to how much over full range you can go. Suppose the meter can accurately integrate spikes of 4X full range--that still leaves you using the 10mA range of a DMM, which will struggle to measure your 4uA residual. And on the 8846A, for example, it would be hopeless as the 10mA range is very noisy.
So assuming you don't find some off-the-shelf solution that I don't know about, the way to go would be analog integration outside the DMM. This may be as simple as an external capacitor across the inputs as has already been mentioned, but I would recommend an external shunt. I'd also consider getting a high-quality polypropylene film cap so that leakage and linearity are not in question. What you need to figure out is how many of those spikes there are and how long they are on average, so you can have some idea what average current will be. Then you can select a current range or shunt and voltage range not too much higher than that and a capacitor large enough so that the maximum signal presented to the DMM doesn't go over range, or at least not too much over range.
The result can then be digitized over a 10 minute period if your DMM supports that. You can also just measure the current with a long aperture, but there will still be dead time between apertures. Whether that is an issue depends on how often the spikes occur and how variable they are in length. Whether all of this passes muster as a 'certified' or 'calibrated' test is another issue. Certainly the DMM and shunt can be calibrated and the precise value of the capacitor doesn't matter, but if someone is looking over your shoulder or demanding paperwork, you may have some convincing to do.