Offstage exterior "bleedthru" sounds I believe are one of the great challenges of backgrounds - up there with wind, ocean/beach textures, and rain. In terms of how to paint it sonically so it pops, but doesn't draw unwanted attention to itself, while also not sounded like a muddy mess of frequencies with a lack of clear intent.
The hallmark to doing this kind of "bleedthru" work, in my opinion, begins with properly establishing the correct type of roomtone for the space. Roomtone may seem like one of those "ah, roomtone, whatever" type sounds - something you just quickly slap in. I think it couldn't be further from the truth. Often in a scene I spend time determining what the space should sound like even before painting in 'textures of life'. No amount of processing or manipulating of bleedthru backgrounds can compensate for having established an inappropriate roomtone. How big is the space? Is it reverberant (like a warehouse or rotunda, or dampened like a bedroom)? What sort of airflow does it have (dead air with a slight clean hiss versus an HVAC vent system, wherein you might hear a light hum or lower frequency content and less of the "quiet' hiss)? How techy of a room is it? Could there be fluorescent buzzes or compressors or fans droning? Whenever I'm establishing a space, even a subjective space like a sci fi operations control room or interrogation room or a generic realistic space, these are the questions I always ask myself before laying in anything - questions I ask while audition source material. Searching, auditioning until I find the textures which hit my ear right. Sometimes it only takes 2 tracks to achieve the sound I want for a roomtone, sometimes it can take 5-6 (believe it or not a 'dead quiet' European opera house I recently cut, pre-performance, commanded about 7 particular roomtone layers to achieve just the right kinds of pleasing, crisp, spacious, cavernous, and dead quiet sound I wanted). But I feel that this is a very important first step.
Once you've determined this, the bleedthru stuff should play very low - often so it's just barely tickling the ear over the roomtone (which is why the proper roomtone is so important!) The roomtone serves almost as your "calibration", so if you have incorrect roomtone, which results in either hearing your bleedthru sounds too crisply/naked, or having to boost them to sound too present in order to cut through the roomtone, then all you're doing is working against yourself.
98% of the time I use exterior-recorded sounds for bleedthru elements - occasionally applying a touch of reverb if needed, but often I leave this up top the stage to handle. I almost NEVER do any EQ (except for notch filtering a highly-offending noise band or doing a roll off, such as unnecessary low end rumble on a bird track) Seeking exterior sounds which have a natural distance and reverb can help, but I even use clean, nearby sounds without much trouble and don't find it bothersome in selling a bleedthru. It comes down to developing a sensibility for what "feels" right in terms of what source material you choose, but also the balance of your bleedthru sounds against your roomtone, the latter of which needs to be appropriate from the start. Whenever I strike this balance, what I hear, even clean ext elements, sounds natural to me and always have.
The stage will usually add the worldizing factor of where the elements get panned within the sound field and what type of a general reverb the BGz will be run through. But I have experienced times where no reverb treatments are used and you know what, the clean sounds,, just tickling the ear over the roomtone, still sound naturally to me and I buy what I'm hearing.
So these may be some things to consider, as well as not thinking too hard over this. BGz are exaggerated for a reason in film/TV, they don't sound right when played as reality would have it - reality is very boring-sounding.