ISM band rules for spread-spectrum: how much do I need to spread?
Treat this answer with appropriate caution, as I have not read the FCC rules for ISM, I am merely reasoning from the information you have provided.

The first extract does not apply to spread spectrum per se, but any form of digital modulation. The difference between SS (assuming direct sequence spread speutrnum DSSS rather than frequency hopping FH, which is more difficult to use) and direct modulation is fundamentally a conceptual one. It is the bit or chip rate that will define the bandwidth of the transmitted channel. It is up to you how you handle the received signal whether the actual data payload is comparable to the bit rate, or much less than it.

You are asked to have a 6dB bandwidth of at least 500kHz. Assume root raised cosine modulation with a data rate of 500k symbols/s. This will have a 3dB bandwidth of 500kHz, which will more than meet that. Whether all of those symbols are independent, or spread by some form of spreading code is up to you.

The first extract does not put other masks on the channel bandwidth, for instance the 99% power is a common one to specify. If this is indeed unspecified, then for RRC filtering you could choose a large \$ \alpha \$ which will simplify your modulation filtering.

The second extract quotes power in a 3kHz bandwidth. This is a very small bandwidth, and if you must have a 500kHz wide channel, 8dBm in every 3kHz would result in a transmitted power of +30dBm. I don't know whether you could get 1 watt in a CubeSat volume, perhaps bursty transmission would be possible.

Anyhow, I think the 3kHz limit is more to prevent 'line like' transmissions. For instance, if you had an IQ modulator, with -20dB carrier leak, and +30dBm output power, you would have +10dBm power being radiated in the DC leak line, which would break this spec. While it should be straightforward to achieve better than 20dB carrier ratio, it may be better to design the modulation without any DC component, so you can AC couple the baseband to the modulator. Modern comms systems do this, 4G's mobile uplink avoids carriers at and near DC for instance.

Having said that how you treat the modulation is up to you, in order to meet the 3kHz power restriction, it must be noise-like. With DSSS, this tends to happen automatically if you spread with m sequences and the like, for data at that rate you will need to use some whitening polynomial to avoid long runs of 0s or 1s that would create discrete looking signals.

I wonder if you can just press some IS-95 mobile radio Qualcomm chips into service? It would save a lot of engineering!