Decompose a number with bit-xor without the digits 0, 3, 7
Common Lisp, 240 224 183 173 169 bytes

Common Lisp is a bit verbose for golfing. However, this decomposes 100-digit numbers in under a second, and 200-digit integers in less than ten seconds, so no need for penalties. The algorithm is deterministic.

(defun s(z)(and #1=(some(lambda(q)(position q(format()"~a"z)))"037")(+ z(floor z(expt 10 #1#)))))
(defun d(x)(do((y x(or(s y)(s #3=(logxor x y))(return`(,y,#3#)))))(())))


The line feed between the functions is for typographic purposes only. Test run with the 100-digit reference input:

(time (d 5296080632396965608312971217160142474083606142654386510789497504098664630388377556711796340247136376))
took 677,000 microseconds (0.677000 seconds) to run.
      20,989 microseconds (0.020989 seconds, 3.10%) of which was spent in GC.
During that period, and with 8 available CPU cores,
     671,875 microseconds (0.671875 seconds) were spent in user mode
           0 microseconds (0.000000 seconds) were spent in system mode
 54,221,104 bytes of memory allocated.
(1864921261592819619661568919418981552559955289196969112566252282429216186594265918444566258544614425
 5891958562486995519825158818455999516899524658151445485616155916296966645869599949958954491929662561)


As a bonus, I'm including a version of the code that incrementally builds the solution top-down. It can manage a 1000-digit number in less than ten seconds, but cannot compete in golf due to the additional code.

(defun decompose (x)
  (flet ((s (z)
           (mapcan #'(lambda (c) (and #1=(position c #2=(format () "~a" z))
                                 (list (- (length #2#) #1# 1))))
                   '(#\0 #\3 #\7))))
    (do ((y x (let ((p (nconc (s y) (s #3=(logxor x y)))))
                (or p (return`(,y,#3#)))
                (+ y (expt 10 (apply #'max p))))))
        (nil))))

* (time (decompose (parse-integer (make-string 1000 :initial-element #\7))))
took 9,226,000 microseconds (9.226000 seconds) to run.
        90,966 microseconds (0.090966 seconds, 0.99%) of which was spent in GC.
During that period, and with 8 available CPU cores,
     9,234,375 microseconds (9.234375 seconds) were spent in user mode
             0 microseconds (0.000000 seconds) were spent in system mode
 487,434,560 bytes of memory allocated.
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
 4184469818464841952189561886965821566229261221619858498284264289194458622668559698924621446851546256444641488616184155821914881485164244662156846141894655485889656891849662551896595944656451462198891289692696856414192264846811616261884188919426294584158925218559295881946496911489245664261126565546419851585441144861859822815144162828551969425529258169849412525611662488849586554989254181228254465226521648916188265491499166186964881248156451994924294646681548996645996894665198811511522424996844864211629888924642289925565591484541149414914699289441561496451494562955652129199261462268846144518142486845251946444998812988291119592418684842524648484689261441456645518518812265495165189812912919529151991611962525419626921619824496626511954895189658691229655648659252448158451924925658586522262194585891859285841914968868466462442488528641466655911199816288496111884591648442984864269495264612518852292965985888414945855422266658614684922884216851481646226111486498155591649619266595911992489425412191)
* (apply #'logxor *)
7777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777