the U5 is a great sounding DI, but def has more of a extended-low end "hifi" kind of sound that works great in some genre's but not necessarily for a vintage flatwound kind of sound. If you can find a Rupert Neve RDI and can deal with only powering it with phantom power that is a very quiet great sounding neutral DI. The API is more in your face and has a great EQ, but the DI is line level not mic level, and is not the easiest thing to deal with. for that kind of $$ you may want to look at some of the tube bass preamps that have a DI out, or the excellent Effectrod LA-1A compressor (that has a transformer base DI output as well, and sounds amazing - i just got one and am blown away). good luck!
Never heard the API but own the Neve RNDI. I wouldn't call it neutral. It has that transformer sound. At home and for recording purposes, I use mostly an UA 610 MKII (tube preamp and optic comp) and a combination of a WA tonebeast preamp, WA76 comp, a Klark LA2A comp and a WA EQP. I'm still curious about the API, 'cause I wouldn't use above mentioned gear live. Live I use a Cali76 comp and just my amp. But I am satisfied so far.
I think you don't do the API justice. In all fairness, it is a kind of channel strip, not just a DI. It is transformer based and offers a preamp, an EQ and a compressor in pedal format. Line level shouldn't be a problem or did you ever find yourself in a situation, where the PA guy told you that they don't have any line level inputs left on their mixer?
But how often do amateurs don't get that a DI box has a mic level output? This might have led to the decision to make it line level. Nothing wrong with that.
@iPinkFender
I have flats on nearly all of my P and J basses. I only play roundwounds on sundays or when I'm drunk

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For me part of the game are your fingers or plectrum and to dial the tone back on passive Fenders and - on a Jazz - to never use both pickups at full volume AND dial the tone back. I've found that preamps are important but overrated. My experience: when I bought the WA76 comp, for the first time in my bass playing live I felt like I sound like I want to sound, like "on a record". The second most important gear was the cheap copy of a LA2A comp from Klark Teknik.
The "classic" bass recording chain is a fast 1176 like comp followed by a slow LA2A like optic comp. In my(!) experience, the impact of these comps is far more important than preamps or DIs.
It is hard for me - to be honest - to hear the differences between different preamps and DIs, but: after owning a lot of cheap comps, esp. pedals - it is not hard for me at all to hear the difference between a quality 1176 comp or a LA2a - and most pedals offered. The Cali76 - as a pedal - was a mindchanger for me, too. Sold all my other pedal-compressors after buying it and use it on every gig since years.