As a former fundamentalist, I can tell you I “came to terms” with it because the doctrine of the Catholic Church is scriptural.
Purgatory- What it is and what it isn’t. It ISN’T a second chance at heaven. When you die you are destined for heaven or hell. Not a second chance for repentance.
What it is?
1030 All who die in God’s grace and friendship, but still imperfectly purified, are indeed assured of their eternal salvation; but after death they undergo purification, so as to achieve the holiness necessary to enter the joy of heaven.
1031 The Church gives the name *Purgatory *to this final purification of the elect, which is entirely different from the punishment of the damned.606 The Church formulated her doctrine of faith on Purgatory especially at the Councils of Florence and Trent. The tradition of the Church, by reference to certain texts of Scripture, speaks of a cleansing fire:607
As for certain lesser faults, we must believe that, before the Final Judgment, there is a purifying fire. He who is truth says that whoever utters blasphemy against the Holy Spirit will be pardoned neither in this age nor in the age to come. From this sentence we understand that certain offenses can be forgiven in this age, but certain others in the age to come.608
1032 This teaching is also based on the practice of prayer for the dead, already mentioned in Sacred Scripture: "Therefore [Judas Maccabeus] made atonement for the dead, that they might be delivered from their sin."609 From the beginning the Church has honored the memory of the dead and offered prayers in suffrage for them, above all the Eucharistic sacrifice, so that, thus purified, they may attain the beatific vision of God.610 The Church also commends almsgiving, indulgences, and works of penance undertaken on behalf of the dead:
Let us help and commemorate them. If Job’s sons were purified by their father’s sacrifice, why would we doubt that our offerings for the dead bring them some consolation? Let us not hesitate to help those who have died and to offer our prayers for them.611 606 Cf. Council of Florence (1439)

S 1304; Council of Trent (1563)

S 1820; (1547):1580; see also Benedict XII,
Benedictus Deus (1336)

S 1000.
607 Cf.
1 Cor 3:15;
1 Pet 1:7.
608 St. Gregory the Great,
Dial. 4,39

L 77,396; cf.
Mt 12:31.
609
2 Macc 12:46.
610 Cf. Council of Lyons II (1274)

S 856.
611 St. John Chrysostom,
Hom. in 1 Cor. 41,5

G 61,361; cf.
Job 1:5
If you believe you are perfect in heaven with no desire to sin, but aren’t perfect yet even though you love Christ and are saved by His blood, you believe in purgatory. Catholics have a name for the change that takes place through CHrist’s saving blood. We call it purgatory. Read the bible references
Indulgences - There are some great little “primers” about indulgences too. I will just put the links in, since I don’t know what “part” of indulgences you think are unbiblical. What people THINK indulgences are is unbiblical, the ABUSE of indulgences was unbiblical, but indulgences themselves are not.
Myths about Indulgences - Read to the end and you will see the scripture quoted. On Catholic Answers website. Explains what it ISN’T.
Primer on Indulgences - Explains what it IS. Scripture throughout.
More if you want, from the Catholic Answers search
As for no salvation outside of the Church, it is true, ONCE you understand that we believe that Christ IS the Church. We can ONLY be saved through Christ and if, as we believe, Christ is the Church, a person could only be saved through the Church.
Christ = the Church
Only way to heaven through Christ
Only way to heaven is through the Church
There is LOTS of teaching on this, again, with scripture.
Here is a link to an article that might help, again, lots of scripture.
No Salvation Outside the Church
I have often said on this said if you show me one doctrine of the CAtholic Church that contradicts scripture, I will leave it forever. I am still here. The Catholic Church is more “biblical” than any “bible believing” church I have ever been to, or studied.
You don’t have to give up you firm belief in scripture. You just need to understand the proper interpretation of it.
God Bless,
Maria