I'm not familiar with many SWL-type receivers , but have been in amateur radio for about 15 years. The primary things I would look for are:
1. The ability to attach an external antenna
2. Some type of narrow filters - adjustable would be awesome but most are switched on or off. The filters should function in AM or SSB - different bandwidths and different filters.
3. Attenuator to reduce the level of strong signals on adjacent frequencies that bleed over and drown out the signal you're trying to rx. This combined with narrow filters can mean the difference between copying something and not.
4. Audio out for external speaker/headphones and computer soundcard connection. There are some software programs that make use of the dsp in your computer soundcard to receive and copy NOAA weather fax images and other info. Now, I've never copied any weather faxes myself, but have used the software for other modes of communications and it's very reliable and very cheap. The software is generally free and a simple interface to isolate the radio from the computer is preferred.
5. Headphones, the Howard Cosell kind that cover your ears. Not expensive Bose noise reducers, but preferably one that is designed for communications and not music. I use Heil, but mainly for the microphone.
6. Less important in a portable- external power.
If portability weren't a priority, I would suggest an older amateur radio, not a vacuum tube boat anchor, but something from at least the late 1980's. There are some fine tabletop receivers out there, but the prices are pretty high. You can find a lot of amateur transceivers with good receivers and the ability to add filters or have DSP filters. Even low-end transceivers will have many of the features like filters and attenuators or the ability to add them. Filters for specific radios are typically around $100, so it could break the budget even if you were willing to go with a larger, older rig. I found
this for $300 on a site where I've purchased many used radios. This is a current model and I've actually owned one. The DSP is pretty poor, but was reliable and filters were available. I used it for digital soundcard modes and emergency backup to another rig. It usually sells for around $450-$500 used. Good luck and I hope I pointed out some things that are useful for you.