Fedi EA got back to me within the hour! Very impressed. Emailing them direct therefore works (for Eurao members).
I missed one bit - the address where I’ll be staying (even if it’s not for the whole duration). I’ll get that firmed up, then all I need to do is activate some summits (and pray for typically dry sunny weather).
My authorisation has come through - it’s official.
It’s best to go down the Fedi EA (Eurao) route to keep things simple - they are very helpful, and membership includes other benefits.
Applications can be sent directly to the ministry - but a digital ID system is in place (for correspondence with government bodies). You need a certificate/ID to access this, but if you’re not a resident, you might not have the necessary digital credentials to access their systems.
Fedi EA will take care of all that. 
Temporary amateur radio license for non-resident foreigners.
In accordance with the provisions of the current Regulations for the Use of the Radio Spectrum by Amateur Radio Operators, approved by Order IET/1311/2013 of July 9, the use of the following amateur radio station is authorized:
• AMATEUR RADIO STATION:
• EQUIPMENT (MAKE, MODEL, SERIAL NUMBER and POWER (W)):
• CALL SIGN:
• AUTHORIZED PERIOD: from _ to _
• OPERATING FREQUENCIES: Those indicated in the attached annex.
• PURPOSE: Transmissions between amateur radio stations, in accordance with the regulations established in the Regulations for the Use of the Radio Spectrum by Amateur Radio Operators.
The amateur radio operator must operate under the limitations and conditions of their license from their country of origin (regarding frequency bands or maximum transmission power), while also complying with the conditions applicable in Spain as outlined in the attached annex (for example, if transmission on a specific frequency band is authorized in Spain, but the foreign amateur radio operator’s license does not include that band, the operator may not transmit on that band even though it is authorized in Spain).
The equipment used must comply with Spanish regulations regarding equipment and devices.
@M9OMS. Nice! 
I look forward to chasing you as an EA5 soon .
73, Simon
I wonder why they have that rule?
It sounds like the older CEPT rules where what you got was the INNER JOIN of your home regs and the host regs.
i.e. UK has no 222MHz and USA does. Me in USA under old CEPT meant no 222MHz. But they got rid of that some years back.
I guess 60m at 1000w is off the cards then.
This is good to know! I’m planning to sit my full exam before my next visit anyway, but my partner’s Spanish, and we usually visit a couple of times a year to see her family. I’d just assumed I wouldn’t be allowed to operate on my intermediate licence. Now I just need to tidy up my dodgy Spanish!
And that is the reason why you must do it!
In my opinion, it will always be simpler for you to operate from Gibraltar than for the rest of the world. 