Personally, I* pronounce this /had had/, with equal and nonzero amounts of stress on both words (assuming I'm not trying to inject emphasis via inflection). There's a stop between them, as two consecutive stressed syllables often don't flow well in English.
So even natives would have made a less hard pronunciation skill, I think.
I don't think this follows. You find /a/ difficult to produce because it's not a sound in your native language, which is perfectly reasonable. But native English speakers don't have this problem; /a/ is a common sound. There is the contracted form of had which makes it simpler to say and sees more use in spoken colloquial English, and while this does eliminate pronunciation of a strong vowel (reduction to a schwa has the same effect but to a lesser degree), I don't think the contraction was born from a desire to avoid saying /had had/.
Consider this sentence which has many consecutive /a/ syllables all bearing the same stress level:
He has had bad, sad anteaters.
A native speaker might have trouble saying this quickly and repeatedly (i.e. it may be a serviceable tongue twister), but in normal speech this wouldn't present any difficulties. I wonder if you're trying to speak too quickly; try saying the above slowly and with a clean stop between each word. That will give you time to form the repeated /a/, hopefully making it less difficult, and also practice saying it (which, as we say in English, makes perfect).
All that said, I don't think it's terribly uncommon (though I also think it's neither usual nor technically correct) to soften either one (but not both, that's definitely wrong) /had/ to /həd/. Use of the schwa and unstressed syllables generally go hand in hand and are extremely common in English. Even when the vowel is properly spoken as a different sound, if we're de-stressing the syllable for some reason, it may well be reduced to a schwa. Contextual emphasis, accent and the fact that stress is somewhat malleable account for this. I'm afraid I can't give you a recommendation on which had is the better candidate for schwa replacement, though.
At the end of the day, if you end up saying had had with a schwa because it's a lot easier for you to say, people will still (most likely) understand you. Better to pronounce something marginally imperfectly than struggle to get it out. In terms of clarity of your communication, it would be best to use the contraction to avoid the double /a/. But I've taken the heart of your question to be about the actual pronunciation, rather than the easiest way to get had had across in speech.
* Information about my accent: I grew up natively speaking American midwestern English. However, years working with (often subpar) voice recognition software and being in British English speaking places have muddled it considerably. There are words I noticeably pronounce "wrong" (even if my accent otherwise fits) among people from both my place of birth and current home. However, I'm reliably informed that my accent is usually still readily identifiable as American (though not as belonging to a particular region).