Xe sources really aren't too bad. I don't think you will find anything that is laser based, provides many wavelengths, and is economic. Look up supercontinuum laser sources as a starting place. Otherwise, you will have to buy individual lasers for each wavelength.

In many fluorescence spectrometers, the old-fashioned, once conventional cw Xe lamps have now bee replaced by Xe short arc flash lamps. These have a similar spectrum, virtually no warm-up time, almost no intensity drift and a long lifetime (10^9 to 10^11 flashes). In the end it depends on the optical setup to provide enough light intensity at the sample... The PerkinElmer LS-50 fluorescence spectrometer was on of the earliest spectrometers using a Xe flash lamp... Provided by mainly two companies: Hamamatsu and Excelitas (once PerkinElmer LifeSciences, once EG&G).

 But still there are Fluorometers with cw XE-Lamps, too. E.g. Fluorolog-3 (Horiba) or FLS-980 (Edinburgh Instruments), of course these instruments are commonly not used in routine labs. Especcially while looking at NIR, an cw arc lamp may be a good idea, but also in VIS. 

Of course there are, why not? I just don't see any advantage from using a cw Xe lamp... They might be much brighter (and hotter), but the important parameter is luminance, where the potential advantage is shrinking away...

Sorry, no... When it comes to commercial product development, new technology gets not necessarily integrated, just because it exists. There are many, many restrictions during product development, and one of the biggest is "never change a running system". It takes time and money and effort and nerves to introduce the slightest product change,  the easiest is always to keep it like it was. To sell this non-impovements then is the marketing department's duty. Believe me, I worked over 10 years in industrial R&D and product development of spectroscopy-based measurement systems.

Technically speaking: no need to lock-in with a flash lamp. Photon-counting is possible,. Flash-to-flash variation might be even smaller than cw fluctuations, excitation intensity is monitored anyways.  High-end spectroscopy is definitely possible (and done) with flash lamps. Brightness is nice, but it depends on luminance and etendue. You can't squeeze light into an area smaller than the light emitting area. So, the main restriction is getting the light through the monochromator slits, where a small emitting area allows to get more light through than from a much brighter bigger emitting area...

Yes, there is a 3rd competitor beside these two instruments using a flash lamp. But in my opinion fluctuation of a cw lamp are less than a flash lamp variations (may be that you can stabilise it, as YAG-Laser manufacturer do).

Back to question: to prevent you from reinventing the wheel- you certainly know these papers?

https://www.researchgate.net/publication/276066057_Determination_of_Photoluminescence_Quantum_Yields_of_Scattering_Media_with_an_Integrating_Sphere_Direct_and_Indirect_Illumination

https://www.researchgate.net/publication/221745212_Integrating_Sphere_Setup_for_the_Traceable_Measurement_of_Absolute_Photoluminescence_Quantum_Yields_in_the_Near_Infrared

https://www.researchgate.net/publication/51032886_Comparison_of_Methods_and_Achievable_Uncertainties_for_the_Relative_and_Absolute_Measurement_of_Photoluminescence_Quantum_Yields

Article Determination of Photoluminescence Quantum Yields of Scatter...

Article Integrating Sphere Setup for the Traceable Measurement of Ab...

Article Comparison of Methods and Achievable Uncertainties for the R...