If you can't buy a specific bub that fits and you don't mind a slightly strange appearance then almost any mains powered dimmable LED bulb that fits and has the correct wattage (that narrow the field :-) ). could have a modifed base added by a competent technician or engineer. Be careful re "dimmable" - some LED bulbs are, some aren't. Some can use standard phase control dimmers, while others can be dimmed only by using a custom control unit.
When replacing a 75 Watt incandescent you need ABOUT 10 - 15 Watts of LED power input to produce the same total light output (lumens), but as the LED light is well directed in this application, the directionality of the LEDs compared to incandescent bulbs means that you would probably be happy with 5 to 10 Watts of total LED power. As LED lumen/Watt efficiency is going up continually, even 5 Watts can be "very very bright" in many applications. (An incandesecnt bulb may produce 15 lumen/Watt. Good modern LEDs give in excess of 120 l/W and last time I checked the top available-to-buy offering from Cree was managig 200 l/W in some very specific circumstances.). Whole lamp l/W will be less than for the LEDs alone due to diffusers, reflectors, power supply losses and stupid designs, but approaching 100 l/W is common enough and higher is becoming available.
The most basic but easiest method would be to start with LEDs and suitable heat sink mountings and use an external power supply fed through the existing wiring. This would allow the lamp to be very close to standard in appearance but involves non standard playing at the other end of the wiring.
Closest to 'plug in and go" (apart from a correctly fitting lamp) is any of the many many LED lamps now available that meets your spec as above, some bases from old bulbs , some suitable adhesive and time skill and patience. With the correct LED lamp it's literally just a matter of connecting the 2 existing wires to the 2 wires from the bulb.
An important factor in many cases is the LED lamp's CRI (Colour Rendering Index). This is a subjective measurement based on a number of colour samples which tells you how well the LED renders the colours of various items compared to a standard tungsten filament incandescent lamp. A tugsten lamp is defined as having a CRI of 100 and LEds can approach that when great care is taken. Not surprisingly, Philips are among the leaders in making high CRI LED lamps.
A CRI of >= 90 is extremely good and few would notice the difference.
A CRI of 80 is good for most purposes by a degree of colour difference is noticeable by many people. Most tasks can be done without major problems.
A CRI of 70 gives light which definitely gives non-tungsten results. These are acceptable in many applications but the difference is clear. eg you'd happily eat a meal, read a book, do homework etc. Colours would be "off" but this would not generally matter except when the colour was a significant part of the task eg looking at photos.
A CRI of 60 is OK for area illumination where colour is not at all important. Camping, tyre changing, night time sports, market stalls (colour suffers), ...
A CRI of under 60 will usually be an actinic blue tinged light pretending to be White. Such LEds are also usually low quality and low lifetime. Their advantages are high light output per power input, all else being equal. Don't buy them!
Hand in hand with CRI is CCT (correlated colo[u]r temperature - but if you get the CRI that you want then CCT will follow. CCT is expressed in degrees Kelvin (same size degree steps as Celsius but starting at absolute zero of = -273 C.) Tungsten has a CCT of about 2900 K, Halogen 3200K, sunlight 5600 K, shade about 7500K. To get a CRI of near 100, CCT will be around 2800K. A CCT of about 6000K is getting down to a CRI of about 70. I personally prefer light of about 6000-7000 K to that of tungsten - slightly on the shady side of noon sunlight. Some love the horrible orange yukky look of tungsten bulbs :-).
See pictures at end.
Two crucial points are:
LED lifetimes are greatly affected by temperature. Keeping them cool is necessary if their days are to be long on the face of the land. Much effort is put into keeping the LEDs at an OK temperature when used in a standard light fitting. If the fitting you are using has obstructed or limited air flow (which is very likely to be the case) then you need to ensure that 'bulb' temperature stays below manufacturers maximum rating. This may required forced air (hidden small fan), or extra metal heat sinking, or running more bulbs than you need at less than full power per bulb. If they share a common air space then having multiple bulbs may still result in excessive temperature.
Brand matters. Y' don't buy no ugly truck!.
If you want good lifetime it pays to ensure that the LEDs proper are made by a master in the dark arts, or has licenced technology from one such. No name no provenance (usually) Asian LEDs can be very good but, alas, more often than not will go down into the grave at a faster rate than is nice. There are industry standard certifications which such products are required to meet to stop the use of junk. I don'y know how well such schemes are policed and certified at the US end, but I do know from extensive general experience at the Asian end of things, that many certifications (CE, ROHS, UL, ....) are quite commonly no more than a printed label. I'm in NZ and have NO factory in China (or anywhere else) and have received invoices in exchange for which the company involved will provide certification that they have visited my factories and that I am doing things as they should be done.
An incomplete list of masters of the arcane art of long life LED design includes
Philips (lumileds, luxeon) - possibly the world's leading exponents of the art.
(Founded 1891. Named '... Philips' Gloeilampenfabrieken' from 1912-1991 and now again at industry front of "Gloeilampen" manufacture.)
Nichia (inventors of the white phosphor LED),
Cree (US face, Asian fab minly). They also sell their LED die to 3rd party encapsulators. Hopefully they do technology transfer of what it trakes to do good encapsulation as well as encapsulation and lead frames are crucial to longevity.
HP / Avago (who do all things well as long as it's not the printer/scanner side of the busines)
and perhaps a few more independently PLUS all their cross licencees.
If in doubt, buy Philips in one of its guises.
Full disclosure: I have no financial interest in Philips or any other LED maker (sadly). I'm just impressed over the decades wity how well Philips does its stuff.
CCT and colour :
From Wikipedia - color temperature
Anywhere on the black curved line (The "Plankian locus") is seen by the eye as white !!!
CCTs are shown on the curved line. Daylight at midday is about central and tungsten is towards the yukky orange end :-).