For maximum altitude, the primary distinction to be made is whether the airplane flies at subsonic or supersonic speed.
Subsonic speed
There are several factors for maximum altitude, but the maximum altitude of specially developed high altitude airplanes is determined by the coffin corner which in turn depends on the product of lift coefficient and Mach number squared. With a given $c_L\cdot M^2$ of about 0.4, which is as good as it gets, now altitude only depends on wing loading which has to be as low as possible for the best altitude. Note that the best subsonic performances were reached with electric propulsion because combustion engines will not work anymore with the low partial pressure of oxygen at altitude.
A high $c_L\cdot M^2$ can best be achieved with generous rear loading on the airfoil. This will upset the longitudinal stability of flying wings - using rear loading at the center will require generous downforce at the tips, and it is the overall $c_L$ that has to be counted for $c_L\cdot M^2$. Reflex airfoils are poor in that respect and do not come close to the value of 0.4 for $c_L\cdot M^2$ which is easily possible with rear loading.
That the altitude record holder is a flying wing shows that low wing loading is the most important feature for high altitude, but the Helios had neither crew nor a serious payload, so its practical usability was very limited. Add to that the fact that the structure was too fragile to survive even moderate gusts and it becomes obvious that a super lightweight span loader is not a practical design.
Supersonic speed
High means fast, and fast means draggy. As can be seen from the envelope of successful designs, they need to fly in a small range of dynamic pressure and have to fly faster to climb higher. Now thermal peak loads will limit what is possible and electric propulsion is out of the question. As a rough estimate for the possible drag at supersonic speed, Dietrich Küchemann once gave this approximation:
$$\left(\frac{L}{D}\right)_{max} = \frac{4\cdot(M+3)}{M}$$ which can be improved for waverider configurations to:
$$\left(\frac{L}{D}\right)_{max} = \frac{6\cdot(M+2)}{M}$$
Regardless which version you choose, both show that L/D suffers with increased flight Mach number $M$. At Mach 5 your best hope for L/D would be 8, only bit more than what Concorde reached at Mach 2. Partial oxygen pressure is of no concern since recompression at the intake will raise the pressure sufficiently - now flow speed and gas temperature in the combustion chamber will set limits to what is possible. The high level of thrust also means that the engine will determine the configuration. Sustained flight at more than Mach 5 has so far only been possible with rockets.
This means your optimum configuration is a slender Delta wing with a lot of internal volume for the hydrogen fuel required by the scramjet or it is a cylindrical pressure vessel with fins.
