Congratulations on your accomplishment! 5 grand is a huge chunk of change to spend, I don't think my recommendations are gonna blow the entire budget. Considering that you mentioned the Oppo HA2-SE, I'm assuming that your source is digital from a computer or some form of transport. Also, you mentioning that you are looking for an alternative to a speaker system due to space constraints therefore I'm gonna suggest open backed headphones.
DAC/AMP
Fostex HPA8 MKII (DAC/AMP); One of the best DACs for extracting detail from sources, and very flexible IO. I love the button in from which allows quick switching from RCA out in the back and the dual 1/4" outs in front if you want to hook small computer speakers to it. Downsides for it though is that it's super heavy therefore not very transportable and the DAC section doesn't work if you don't hook it up to the mains.
Ressonance Labs Concero HD (DAC only); This one is more musical and is powered by USB and has a very small footprint. Doesn't have the flexibility of most DACs at this price range as it only has one set of RCA outs and USB/COAX inputs.
SPL series of Phonitor amps (AMP); All requires mains power. They have the best crossfeed function and the bigger (more expensive) the amp the more ball and whistles you have, for example, a balanced output. This would replicate the "speaker system sound" that you're striving for. The entire line of amps can power all headphones that I know of, discounting electrostatics.
Headphones
As I have recommended solid state amps, for detailed extraction of musicality, I'm going to suggest laid back headphones to achieve the relaxed to calm the natural analytical nature of solid states.
Audeze LCD3; I love it for the accurate bass and detailed treble. Not too difficult to drive if you so decide to switch it up down the line with different amps. This however has a V shaped sound signature there vocals would be slightly recessed. For long listening sessions, I find this headphone a little heavy, 2 hours maximum for me, creating a hot spot where it rests on top of my small head. I look ridiculous in it. Oh, also this one would blow approx half your budget.
Focal Elear; Half the price of the LCDs with albeit more punchy low end and dynamics. It's sound signature is similar to HD650, if you ever auditioned those, but more dynamic and better low end, overall clarity and soundstage. Hands down, one of the most comfortable headphones considering it's weight and size but the only thing annoys me is the long heavy (but quality) cable that comes with it.
Senheisser HD650. I don't think that this headphone needs any explanation. This is my personal go-to-sleep headphones, lightweight and comfortable with a soothing sound signature.
Well, that's my suggestions and I would love to hear what you think of it and ultimately what you decided to get. You'll also have change based on my suggestions where you can spend some of it on upgrade cables for your equipment.